diff --git "a/09AyT4oBgHgl3EQf1PmO/content/tmp_files/load_file.txt" "b/09AyT4oBgHgl3EQf1PmO/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/09AyT4oBgHgl3EQf1PmO/content/tmp_files/load_file.txt" @@ -0,0 +1,925 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf,len=924 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='00732v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='CC] 2 Jan 2023 Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank Dror Chawin* Ishay Haviv* Abstract The orthogonality dimension of a graph G over R is the smallest integer k for which one can assign a nonzero k-dimensional real vector to each vertex of G, such that every two adjacent vertices receive orthogonal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We prove that for every sufficiently large integer k, it is NP-hard to decide whether the orthogonality dimension of a given graph over R is at most k or at least 2(1−o(1))·k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This in particular implies that it is NP-hard to approximate these graph quantities to within any constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their com- plement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1 Introduction A graph G is said to be k-colorable if its vertices can be colored by k colors such that every two ad- jacent vertices receive distinct colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The chromatic number of G, denoted by χ(G), is the smallest integer k for which G is k-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' As a fundamental and popular graph quantity, the chromatic number has received a considerable amount of attention in the literature from a computational perspective, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The problem of deciding whether a graph G satisfies χ(G) ≤ 3 is one of the classical twenty- one NP-complete problems presented by Karp [26] in 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Khanna, Linial, and Safra [28] proved that it is NP-hard to distinguish between graphs G that satisfy χ(G) ≤ 3 from those satisfying χ(G) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This result, combined with the approach of Garey and Johnson [15] and with a result of Stahl [39], implies that for every k ≥ 6, it is NP-hard to decide whether a graph G satisfies χ(G) ≤ k or χ(G) ≥ 2k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Brakensiek and Guruswami [6] proved that for every k ≥ 3, it is NP-hard to distinguish between the cases χ(G) ≤ k and χ(G) ≥ 2k − 1, and the 2k − 1 bound was further improved to 2k by Barto, Bul´ın, Krokhin, and Oprˇsal [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For large values of k, it was shown by Khot [29] that it is NP-hard to decide whether a graph G satisfies χ(G) ≤ k or χ(G) ≥ kΩ(log k), and the latter condition was strengthened to χ(G) ≥ 2k1/3 by Huang [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' A substantial improvement School of Computer Science, The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Research supported by the Israel Science Foundation (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1218/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1 was recently obtained by Wrochna and ˇZivn´y [40], who proved that for every k ≥ 4, it is NP- hard to decide whether a given graph G satisfies χ(G) ≤ k or χ(G) ≥ ( k ⌊k/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The proof of this result combined the hardness result of [24] with the construction of line digraphs [20] and with a result of Poljak and R¨odl [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that under certain variants of the Unique Games Conjecture, stronger hardness results are known to hold, namely, hardness of deciding whether a given graph G satisfies χ(G) ≤ k1 or χ(G) ≥ k2 for all integers k2 > k1 ≥ 3 [10] (see also [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The present paper studies the computational complexity of algebraic variants of the chromatic number of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' A k-dimensional orthogonal representation of a graph G = (V, E) over a field F is an assignment of a vector uv ∈ Fk with ⟨uv, uv⟩ ̸= 0 to each vertex v ∈ V, such that for every two adjacent vertices v and v′ it holds that ⟨uv, uv′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Here, for two vectors x, y ∈ Fk, we consider the standard inner product defined by ⟨x, y⟩ = ∑k i=1 xiyi with operations over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The orthogonality dimension of G over F, denoted by ξF(G), is the smallest integer k for which G admits a k-dimensional orthogonal representation over F (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It can be easily seen that for every graph G and for every field F, it holds that ξF(G) ≤ χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In addition, if F is a fixed finite field or the real field R, it further holds that ξF(G) ≥ Ω(log χ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Both bounds are known to be tight in the worst case (see Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6 and [33, Chapter 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The study of orthogonal representations and orthogonality dimension was initiated in the seminal work of Lov´asz [32] on the ϑ-function and has found applications in various areas, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=', information theory [32], graph theory [34], and quantum communication complexity [9, Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The interest in the hardness of determining the orthogonality dimension of graphs dates back to a paper of Lov´asz, Saks, and Schrijver [34], where it was noted that the problem seems difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The aforementioned relations between the chromatic number and the orthogonality dimension yield that hardness of deciding whether a graph G satisfies χ(G) ≤ k1 or χ(G) ≥ k2 implies the hardness of deciding whether it satisfies ξF(G) ≤ k1 or ξF(G) ≥ Ω(log k2), provided that F is a finite field or R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It therefore follows from [10] that assuming certain variants of the Unique Games Conjecture, it is hard to decide whether a graph G satisfies ξF(G) ≤ k1 or ξF(G) ≥ k2 for all integers k2 > k1 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This reasoning, however, does not yield NP-hardness results for the orthogonality dimension (without additional complexity assumptions), even using the strongest known NP-hardness results of the chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Yet, a result of Peeters [35] implies that for every field F, it is NP-hard to decide if a given graph G satisfies ξF(G) ≤ 3, hence it is NP-hard to approximate the orthogonality dimension of a graph over F to within any factor smaller than 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Over the reals, the hardness of approximation for the orthogonality dimension was recently extended in [16] to any factor smaller than 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Another algebraic quantity of graphs is the minrank parameter that was introduced in 1981 by Haemers [19] in the study of the Shannon capacity of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The minrank parameter was used in [18, 19] to answer questions of Lov´asz [32] and was later applied by Alon [1], with a different formulation, to disprove a conjecture of Shannon [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The minrank of a graph G over a field F, denoted by minrkF(G), is closely related to the orthogonality dimension of the complement graph G over F and satisfies minrkF(G) ≤ ξF(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The difference between the two quantities comes, roughly speaking, from the fact that the definition of minrank involves the notion of orthogonal bi- representations rather than orthogonal representations (for the precise definitions, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The study of the minrank parameter is motivated by various applications in information theory and in theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' A prominent one is the well-studied index coding problem, 2 for which the minrank parameter perfectly characterizes the optimal length of its linear solutions, as was shown by Bar-Yossef, Birk, Jayram, and Kol [3] (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Similarly to the situation of the orthogonality dimension, it was proved in [35] that for every field F, it is NP-hard to decide if a given graph G satisfies minrkF(G) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It was further shown by Dau, Skachek, and Chee [8] that it is NP-hard to decide whether a given digraph G satisfies minrkF2(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that for (undirected) graphs, the minrank over any field is at most 2 if and only if the complement graph is bipartite, a property that can be checked in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Motivated by the computational aspects of the index coding problem, Langberg and Sprintson [30] related the minrank of a graph to the chromatic number of its complement and derived from [10] that assuming certain variants of the Unique Games Conjecture, it is hard to decide whether a given graph G satisfies minrkF(G) ≤ k1 or minrkF(G) ≥ k2, provided that k2 > k1 ≥ 3 and that F is a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Similar hardness results were obtained in [30] for additional settings of the index coding problem, including the general (non-linear) index coding problem over a constant-size alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 Our Contribution This paper provides improved NP-hardness of approximation results for the orthogonality dimen- sion and for the minrank parameter over various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We start with the following result, which is concerned with the orthogonality dimension over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' There exists a function f : N → N satisfying f(k) = 2(1−o(1))·k/2 such that for every sufficiently large integer k, it is NP-hard to decide whether a given graph G satisfies ξR(G) ≤ k or ξR(G) ≥ f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 implies that it is NP-hard to approximate the orthogonality dimension of a graph over the reals to within any constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Previously, such NP-hardness result was known to hold only for approximation factors smaller than 3/2 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We proceed with the following result, which is concerned with the orthogonality dimension and the minrank parameter over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every finite field F, there exists a function f : N → N satisfying f(k) = 2(1−o(1))·k/2 such that for every sufficiently large integer k, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is NP-hard to decide whether a given graph G satisfies ξF(G) ≤ k or ξF(G) ≥ f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is NP-hard to decide whether a given graph G satisfies minrkF(G) ≤ k or minrkF(G) ≥ f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 implies that over any finite field, it is NP-hard to approximate the orthogonality di- mension and the minrank of a graph to within any constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let us stress that this hardness result relies solely on the assumption P ̸= NP rather than on stronger complexity assumptions and thus settles a question raised in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Prior to this work, it was known that it is NP-hard to approximate the minrank of graphs to within any factor smaller than 4/3 [35] and the minrank of digraphs over F2 to within any factor smaller than 3/2 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' A central component of the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 is the notion of line digraphs, introduced in [20], that was first used in the context of hardness of approximation by Wrochna 3 and ˇZivn´y [40] (see also [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It was shown in [21, 36] that the chromatic number of any graph is exponential in the chromatic number of its line digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This result was iteratively applied by the authors of [40] to improve the NP-hardness of the chromatic number from the k vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 2k1/3 gap of [24] to their k vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' ( k ⌊k/2⌋) gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The main technical contribution of the present work lies in analyzing the orthogonality dimension of line digraphs and the minrank parameter of their complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We actually show that on line digraphs, these graph parameters are quadratically related to the chromatic number (see Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This allows us to derive our hardness results from the hardness of the chromatic number given in [40], where the obtained gaps are only quadratically weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We further discuss some limitations of our approach, involving an analogue of Sperner’s theorem for subspaces due to Kalai [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We finally show that our approach might be useful for proving hardness results for the general (non-linear) index coding problem over a constant-size alphabet, for which no NP-hardness result is currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It was shown by Langberg and Sprintson [30] that for an instance of the index coding problem represented by a graph G, the length of an optimal solution is at most χ(G) and at least Ω(log log χ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It thus follows that an NP-hardness result for the chromatic number with a double-exponential gap would imply an NP-hardness result for the general index coding prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' However, no such NP-hardness result is currently known for the chromatic number without relying on further complexity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To tackle this issue, we study the index coding prob- lem on instances which are complement of line digraphs (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' As a consequence of our results, we obtain that the NP-hardness of the general index coding problem can be derived from an NP-hardness result of the chromatic number with only a single-exponential gap, not that far from the best known gap given in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a precise statement, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 Related Work We gather here several related results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' A result of Zuckerman [41] asserts that for any ε > 0, it is NP-hard to approximate the chro- matic number of a graph on n vertices to within a factor of n1−ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It would be interesting to figure out if such hardness result holds for the orthogonality dimension and for the min- rank parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The present paper, however, focuses on the hardness of gap problems with constant thresholds, independent of the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' As mentioned earlier, Peeters [35] proved that for every field F, it is NP-hard to decide if the minrank (or the orthogonality dimension) of a given graph is at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We note that for finite fields, this can also be derived from a result of Hell and Neˇsetˇril [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For the chromatic number of hypergraphs, the gaps for which NP-hardness is known to hold are much stronger than for graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For example, it was shown in [5] that for some δ > 0, it is NP-hard to decide if a given 4-uniform hypergraph G on n vertices satisfies χ(G) ≤ 2 or χ(G) ≥ logδ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' An analogue result for the orthogonality dimension of hypergraphs over R was proved in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' On the algorithmic side, a long line of work has explored the number of colors that an effi- cient algorithm needs for properly coloring a given k-colorable graph, where k ≥ 3 is a fixed 4 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For example, there exists a polynomial-time algorithm that on a given 3-colorable graph with n vertices uses O(n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='19996) colors [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Algorithms of this nature exist for the graph parameters studied in this work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Indeed, there exists a polynomial-time algo- rithm that given a graph G on n vertices with ξR(G) ≤ 3 finds a proper coloring of G with O(n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2413) colors [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Further, there exists a polynomial-time algorithm that given a graph G on n vertices with minrkF2(G) ≤ 3 finds a proper coloring of G with O(n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2574) colors [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that the colorings obtained by these two algorithms provide, respectively, orthogonal and bi-orthogonal representations for the input graph G (see Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='3 Outline The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In Section 2, we collect several definitions and results that will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In Section 3, we study the underlying graphs of line digraphs and their behavior with respect to the orthogonality dimension, the minrank parameter, and the index coding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We also discuss there some limitations of our approach, given in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Finally, in Section 4, we prove our hardness results and complete the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 2 Preliminaries Throughout the paper, undirected graphs are referred to as graphs, and directed graphs are re- ferred to as digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' All the considered graphs and digraphs are simple, and all the logarithms are in base 2 unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For an integer n, we use the notation [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 Orthogonality Dimension and Minrank The orthogonality dimension of a graph is defined as follows (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=', [33, Chapter 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 (Orthogonality Dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' A k-dimensional orthogonal representation of a graph G = (V, E) over a field F is an assignment of a vector uv ∈ Fk with ⟨uv, uv⟩ ̸= 0 to each vertex v ∈ V, such that ⟨uv, uv′⟩ = 0 whenever v and v′ are adjacent vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Here, for two vectors x, y ∈ Fk, we let ⟨x, y⟩ = ∑k i=1 xiyi denote the standard inner product of x and y over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The orthogonality dimension of a graph G over a field F, denoted by ξF(G), is the smallest integer k for which there exists a k-dimensional orthogonal representation of G over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We note that orthogonal representations are sometimes defined in the literature such that the vectors associated with non-adjacent vertices are required to be orthogonal, that is, as orthogonal represen- tations of the complement graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' While we find it more convenient to use the other definition in this paper, one can view the notation ξF(G) as standing for ξF(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=', the orthogonality dimension of the complement graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The same holds for the notion of orthogonal bi-representations, given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The minrank parameter, introduced in [19], is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='3 (Minrank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let G = (V, E) be a digraph on the vertex set V = [n], and let F be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We say that a matrix M ∈ Fn×n represents G if Mi,i ̸= 0 for every i ∈ V, and Mi,j = 0 for every distinct 5 vertices i, j ∈ V such that (i, j) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The minrank of G over F is defined as minrkF(G) = min{rankF(M) | M represents G over F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The definition is naturally extended to graphs by replacing every edge with two oppositely directed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We next describe an alternative definition due to Peeters [35] for the minrank of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This requires the following extension of orthogonal representations, called orthogonal bi-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' A k-dimensional orthogonal bi-representation of a graph G = (V, E) over a field F is an assignment of a pair of vectors (uv, wv) ∈ Fk × Fk with ⟨uv, wv⟩ ̸= 0 to each vertex v ∈ V, such that ⟨uv, wv′⟩ = ⟨uv′, wv⟩ = 0 whenever v and v′ are adjacent vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The following proposition follows directly from Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4 combined with the fact that for every matrix M ∈ Fn×n, rankF(M) is the smallest integer k for which M can be written as M = MT 1 · M2 for two matrices M1, M2 ∈ Fk×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5 ([35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every field F and for every graph G, minrkF(G) is the smallest integer k for which there exists a k-dimensional orthogonal bi-representation of G over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The following claim summarizes some known relations between the studied graph parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We provide a quick proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every field F and for every graph G, it holds that minrkF(G) ≤ ξF(G) ≤ χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In addition, if F is finite, then minrkF(G) ≥ log|F| χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: The inequality minrkF(G) ≤ ξF(G) follows by combining Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5 with the fact that a k-dimensional orthogonal representation of G over F induces a k-dimensional orthogonal bi-representation of G over F with two identical vectors for every vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For the inequality ξF(G) ≤ χ(G), observe that any proper coloring of G with k colors induces a k-dimensional orthogonal representation of G over any field F, by assigning the ith vector of the standard basis of Fk to each vertex colored by the ith color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Next, assuming that F is finite, we show that minrkF(G) ≥ log|F| χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To this end, denote k = minrkF(G), and apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5 to obtain that there exists a k-dimensional orthogonal bi-representation of G over F that assigns a pair (uv, wv) ∈ Fk × Fk to each vertex v of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every two adjacent vertices v and v′ in G, the vectors uv and uv′ are distinct, because ⟨uv, wv′⟩ = 0 whereas ⟨uv′, wv′⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This implies that G admits a proper coloring with at most |F|k colors, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We finally recall that a homomorphism from a graph G1 = (V1, E1) to a graph G2 = (V2, E2) is a function g : V1 → V2 such that for every two vertices x, y ∈ V1 with {x, y} ∈ E1, it holds that {g(x), g(y)} ∈ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Observe that if there exists a homomorphism from G1 to G2 then we have χ(G1) ≤ χ(G2), and for every field F, ξF(G1) ≤ ξF(G2) and minrkF(G1) ≤ minrkF(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 Index Coding The index coding problem, introduced in [3], is concerned with economical strategies for broad- casting information to n receivers in a way that enables each of them to retrieve its own message, a symbol from some given alphabet Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For this purpose, each receiver is allowed to use some prior side information that consists of a subset of the messages required by the other receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The side information map is naturally represented by a digraph on [n], which includes an edge (i, j) if the ith receiver knows the message required by the jth receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The objective is to minimize the length of the transmitted information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For simplicity, we consider here the case of symmetric side information maps, represented by graphs rather than by digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The formal definition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7 (Index Coding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let G be a graph on the vertex set [n], and let Σ be an alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' An index code for G over Σ of length k is an encoding function E : Σn → Σk such that for every i ∈ [n], there exists a decoding function gi : Σk+|NG(i)| → Σ, such that for every x ∈ Σn, it holds that gi(E(x), x|NG(i)) = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Here, NG(i) stands for the set of vertices in G adjacent to the vertex i, and x|NG(i) stands for the restriction of x to the indices of NG(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If Σ is a field F and the encoding function E is linear over F, then we say that the index code is linear over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Bar-Yossef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' [3] showed that the minrank parameter characterizes the length of optimal solutions to the index coding problem in the linear setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='8 ([3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every field F and for every graph G, the minimal length of a linear index code for G over F is minrkF(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 3 Line Digraphs In 1960, Harary and Norman [20] introduced the concept of line digraphs, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 (Line Digraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a digraph G = (V, E), the line digraph of G, denoted by δG, is the digraph on the vertex set E that includes a directed edge from a vertex (x, y) to a vertex (z, w) whenever y = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 is naturally extended to graphs G by replacing every edge of G with two oppositely directed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that in this case, the number of vertices in δG is twice the number of edges in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We will frequently consider the underlying graph of the digraph δG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=', the graph obtained from δG by ignoring the directions of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The following result of Poljak and R¨odl [36], which strengthens a previous result of Harner and Entringer [21], shows that the chromatic number of a graph G precisely determines the chromatic number of the underlying graph of δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The statement of the result uses the function b : N → N defined by b(n) = ( n ⌊n/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 ([21, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let G be a graph, and let H be the underlying graph of the digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, χ(H) = min{n | χ(G) ≤ b(n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Using the fact that b(n) ∼ 2n √ πn/2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 implies that the chromatic number of G is expo- nential in the chromatic number of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Our goal in this section is to relate the chromatic number of G to other graph parameters of H, namely, the orthogonality dimension, the minrank of the complement, and the optimal length of an index code for the complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 Orthogonality Dimension For a field F, an integer n, and a subspace U of Fn, we denote by U⊥ the subspace of Fn that consists of the vectors that are orthogonal to U over F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=', U⊥ = {w ∈ Fn | ⟨w, u⟩ = 0 for every u ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Consider the following family of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a field F and an integer n, let S1(F, n) denote the graph whose vertices are all the subspaces of Fn, where two distinct subspaces U1 and U2 are adjacent if there exists a vector w ∈ Fn with ⟨w, w⟩ ̸= 0 that satisfies w ∈ U1 ∩ U⊥ 2 and, in addition, there exists a vector w′ ∈ Fn with ⟨w′, w′⟩ ̸= 0 that satisfies w′ ∈ U2 ∩ U⊥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In words, two subspaces of Fn are adjacent in the graph S1(F, n) if each of them includes a non- self-orthogonal vector that is orthogonal to the entire other subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that for an infinite field F and for n ≥ 2, the vertex set of S1(F, n) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We argue that the chromatic number of a graph G can be used to estimate the orthogonality dimension of the underlying graph H of its line digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' First, recall that by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2, the chromatic number of H is logarithmic in χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This implies, using Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6, that the orthog- onality dimension of H over any field is at most logarithmic in χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a lower bound on the orthogonality dimension of H, we need the following lemma that involves the chromatic numbers of the graphs S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let F be a field, let G be a graph, let H be the underlying graph of the digraph δG, and put n = ξF(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, χ(G) ≤ χ(S1(F, n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Put G = (VG, EG) and H = (VH, EH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The assumption n = ξF(H) implies that there exists an n-dimensional orthogonal representation of H over F, that is, an assignment of a vector uv ∈ Fn with ⟨uv, uv⟩ ̸= 0 to each vertex v ∈ VH, such that ⟨uv, uv′⟩ = 0 whenever v and v′ are adjacent in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Recall that the vertices of H, just as the vertices of δG, are the ordered pairs (x, y) of adjacent vertices x, y in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every vertex y ∈ VG, let Uy denote the subspace spanned by the vectors of the given orthogonal representation that are associated with the vertices of H whose tail is y, namely, Uy = span({uv | v = (x, y) for some x ∈ VG}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that Uy is a subspace of Fn, and thus a vertex of S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Consider the function that maps every vertex y ∈ VG of G to the vertex Uy of S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We claim that this function forms a homomorphism from G to S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, let x, y ∈ VG be adjacent vertices in G, and consider the vector w = u(x,y) assigned by the given orthogonal representation to the vertex (x, y) of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By the definition of an orthogonal representation, it holds that ⟨w, w⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since (x, y) is a vertex of H whose tail is y, it follows that w ∈ Uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Further, every vertex of H of the form (x′, x) for some x′ ∈ VG is adjacent in H to (x, y), hence it holds that ⟨u(x′,x), w⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since the subspace Ux is spanned by those vectors u(x′,x), we obtain that w is orthogonal to the entire subspace Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It thus follows that the vector w satisfies ⟨w, w⟩ ̸= 0 and w ∈ Uy ∩ U⊥ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By symmetry, there also exists a vector w′ ∈ Fn satisfying ⟨w′, w′⟩ ̸= 0 and w′ ∈ Ux ∩ U⊥ y , hence the subspaces Ux 8 and Uy are adjacent vertices in S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We conclude that the above function is a homomorphism from G to S1(F, n), hence the chromatic numbers of these graphs satisfy χ(G) ≤ χ(S1(F, n)), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In order to derive useful bounds from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4, we need upper bounds on the chromatic numbers of the graphs S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Every vertex of S1(F, n) is a subspace of Fn and thus can be represented by a basis that generates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a finite field F of size q, the number of possible bases does not exceed qn2, which obviously yields that χ(S1(F, n)) ≤ qn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' While this simple bound suffices for proving our hardness results for the orthogonality dimension over finite fields, we note that the number of vertices in S1(F, n) is in fact q(1+o(1))·n2/4, where the o(1) term tends to 0 when n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 We conclude this discussion with the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let F be a finite field of size q, let G be a graph, and let H be the underlying graph of the digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, it holds that ξF(H) ≥ � logq χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Put n = ξF(H), and apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4 to obtain that χ(G) ≤ χ(S1(F, n)) ≤ qn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By rear- ranging, the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 The Chromatic Number of S1(R, n) For the real field R and for n ≥ 2, the vertex set of the graph S1(R, n) is infinite, and yet, its chromatic number is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, let us firstly observe a simple upper bound of 23n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To each vertex of S1(R, n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=', a subspace U of Rn, assign the subset of {0, ±1}n that consists of all the sign vectors of the vectors of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This assignment forms a proper coloring of the graph, because for adjacent vertices U and V there exists a nonzero vector w ∈ U that is orthogonal to V, hence the sign vector of w belongs to the set of sign vectors of U but does not belong to the one of V (because the inner product of two vectors with the same nonzero sign vector is positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since the number of subsets of {0, ±1}n is 23n, it follows that χ(S1(R, n)) ≤ 23n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The above double-exponential bound is not sufficient for deriving NP-hardness of approxima- tion results for the orthogonality dimension over R from the currently known NP-hardness results of the chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We therefore need the following lemma that provides an exponentially better bound which is suitable for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a vector w ∈ Rn, we use here the notation ∥w∥ = � ⟨w, w⟩ for the Euclidean norm of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every integer n, it holds that χ(S1(R, n)) ≤ (2n + 1)n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: We define a coloring of the vertices of the graph S1(R, n) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every vertex of S1(R, n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=', a subspace U of Rn, let (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , uk) be an arbitrary orthonormal basis of U where k ≤ n, and assign U to the color c(U) = (u′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , u′ k) where u′ i is a vector obtained from ui by 1To see this, observe that the number of k-dimensional subspaces of Fn is precisely ∏k−1 i=0 qn−qi qk−qi and that every term in this product lies in [qn−k−1, qn−k+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Hence, the total number of subspaces of Fn is at least ∑n k=0 q(n−k−1)k and at most ∑n k=0 q(n−k+1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It follows that the number of subspaces of Fn is q(1+o(1))·n2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 9 rounding each of its values to a closest integer multiple of 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that for every i ∈ [k], the vectors ui and u′ i differ in every coordinate by no more than 1 2n in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We claim that c is a proper coloring of S1(R, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, let U and V be adjacent vertices in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If dim(U) ̸= dim(V) then it clearly holds that c(U) ̸= c(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' So suppose that the dimensions of U and V are equal, and put k = dim(U) = dim(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Denote the orthonormal bases associated with U and V by (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , uk) and (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , vk) respectively, and let c(U) = (u′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , u′ k) and c(V) = (v′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , v′ k) be their colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Our goal is to show that c(U) ̸= c(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Assume for the sake of contradiction that c(U) = c(V), that is, u′ i = v′ i for every i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This implies that for every i ∈ [k], the vectors ui and vi differ in each coordinate by no more than 1 n in absolute value, hence ∥ui − vi∥ ≤ � n · 1 n2 = 1 √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' (1) Since U and V are adjacent in the graph S1(R, n), by scaling, there exists a unit vector u ∈ U ∩ V⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Write u = ∑i∈[k] αi · ui for coefficients α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , αk ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since the given basis of U is orthonormal, it follows that ∑i∈[k] α2 i = ∥u∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Now, consider the vector v = ∑i∈[k] αi · vi, and observe that v is a unit vector that belongs to the subspace V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Observe further that ∥u − v∥ = ��� ∑ i∈[k] αi · (ui − vi) ��� ≤ ∑ i∈[k] |αi| · ∥ui − vi∥ ≤ � ∑ i∈[k] α2 i �1/2 � ∑ i∈[k] ∥ui − vi∥2�1/2 ≤ 1, (2) where the first inequality follows from the triangle inequality, the second from the Cauchy-Schwarz inequality, and the third from (1) using k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' However, u and v are orthogonal unit vectors, and as such, the distance between them satisfies ∥u − v∥ = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This yields a contradiction to (2), hence c(U) ̸= c(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To complete the proof, we observe that the number of colors used by the proper coloring c does not exceed (2n + 1)n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Indeed, every color can be represented by an n × n matrix whose values are of the form a n for integers −n ≤ a ≤ n (where the matrix associated with a subspace of dimension k consists of the rounded k column vectors concatenated with n − k columns of zeros).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since the number of those matrices is bounded by (2n + 1)n2, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We derive the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' There exists a constant c > 0, such that for every graph G with χ(G) ≥ 3, the underlying graph H of the digraph δG satisfies ξR(H) ≥ c · � log χ(G) log log χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Put n = ξR(H), and combine Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4 with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6 to obtain that χ(G) ≤ χ(S1(R, n)) ≤ (2n + 1)n2, which yields the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 The Clique Number of S1(F, n) We next consider the clique numbers of the graphs S1(F, n), whose estimation is motivated by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Here, the clique number of a graph G is denoted by ω(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let F be a field, let G be a graph, and let H be the underlying graph of the digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If χ(G) ≤ ω(S1(F, n)), then ξF(H) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Put m = ω(S1(F, n)), and let U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , Um be m subspaces of Fn that form a clique in S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Put G = (V, E), suppose that χ(G) ≤ m, and let c : V → [m] be a proper coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Notice that for every two adjacent vertices x, y in G, the subspaces Uc(x) and Uc(y) are adjacent vertices in S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We define an n-dimensional orthogonal representation of H over F as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Recall that every vertex of H is a pair (x, y) of adjacent vertices x, y in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Assign every such vertex (x, y) to some non-self-orthogonal vector u(x,y) that lies in Uc(y) ∩ U⊥ c(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The existence of such a vector follows from the adjacency of the vertices Uc(x) and Uc(y) in S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We claim that this assign- ment is an orthogonal representation of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Indeed, for adjacent vertices (x, y) and (y, z) in H, the vector u(x,y) belongs to Uc(y) whereas the vector u(y,z) is orthogonal to Uc(y), hence they satisfy ⟨u(x,y), u(y,z)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since this orthogonal representation lies in Fn, we establish that ξF(H) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a graph G and for the underlying graph H of its line digraph δG, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 implies that if χ(G) ≤ ( n ⌊n/2⌋) then χ(H) ≤ n, and thus, by Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6, ξF(H) ≤ n for every field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This raises the question of whether Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='8 can be used to obtain a better upper bound on ξF(H) as a function of χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For certain cases, the following result answers this question negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Namely, it shows that the clique number of the graphs S1(F, n) is precisely ( n ⌊n/2⌋), whenever the vector space Fn has no nonzero self-orthogonal vectors (as in the case of F = R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It thus follows that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='8 cannot yield a better relation between the quantities ξR(H) and χ(G) than the one stemming from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a field F and an integer n such that Fn has no nonzero self-orthogonal vectors, it holds that ω(S1(F, n)) = � n ⌊n/2⌋ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='9 relies on the following result of Kalai [25] (see also [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='10 ([25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a field F and an integer n, let (U1, W1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , (Um, Wm) be m pairs of subspaces of Fn such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Ui ∩ Wi = {0} for every i ∈ [m], and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Ui ∩ Wj ̸= {0} for every i ̸= j ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, m ≤ ( n ⌊n/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='9: We first show that there exists a clique in S1(F, n) of size ( n ⌊n/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every set A ⊆ [n] of size |A| = ⌊n/2⌋, let UA denote the subspace of Fn spanned by the vectors ei with i ∈ A, where ei stands for the vector of Fn with 1 on the ith entry and 0 everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 11 It clearly holds that for every distinct such sets A1, A2, there exists some i ∈ A1 \\ A2, and that the vector ei satisfies ⟨ei, ei⟩ = 1 and ei ∈ UA1 ∩ U⊥ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It thus follows that the ( n ⌊n/2⌋) subspaces UA with |A| = ⌊n/2⌋ form a clique in the graph S1(F, n), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We next show that the size of every clique in S1(F, n) does not exceed ( n ⌊n/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, let U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , Um be subspaces of Fn that form a clique in S1(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Consider the pairs (Ui, U⊥ i ) for i ∈ [m], and observe that they satisfy the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Indeed, for every i ∈ [m] it holds that Ui ∩ U⊥ i = {0}, because Fn has no nonzero self-orthogonal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Further, since the given collection of subspaces is a clique in S1(F, n), for every i ̸= j ∈ [m], there exists a vector w ∈ Fn with ⟨w, w⟩ ̸= 0 such that w ∈ Ui ∩ U⊥ j , hence, Ui ∩ U⊥ j ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It thus follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='10 that m ≤ ( n ⌊n/2⌋), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2 Minrank As in the previous section, we start with a definition of a family of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For a field F and an integer n, let S2(F, n) denote the graph whose vertices are all the pairs of subspaces of Fn, where two distinct pairs (U1, W1) and (U2, W2) are adjacent if there exist two vectors u, w ∈ Fn with ⟨u, w⟩ ̸= 0 such that u ∈ U1 ∩ W⊥ 2 and w ∈ W1 ∩ U⊥ 2 and, in addition, there exist two vectors u′, w′ ∈ Fn with ⟨u′, w′⟩ ̸= 0 such that u′ ∈ U2 ∩ W⊥ 1 and w′ ∈ W2 ∩ U⊥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We next argue that the chromatic number of a graph G can be used to estimate the minrank of the complement of the underlying graph of its line digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This is established using the following lemma that involves the chromatic numbers of the graphs S2(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Its proof resembles that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let F be a field, let G be a graph, let H be the underlying graph of the digraph δG, and put n = minrkF(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, χ(G) ≤ χ(S2(F, n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Put G = (VG, EG) and H = (VH, EH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The assumption n = minrkF(H) implies, by Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5, that there exists an n-dimensional orthogonal bi-representation of H over F, that is, an assignment of a pair of vectors (uv, wv) ∈ Fn × Fn with ⟨uv, wv⟩ ̸= 0 to each vertex v ∈ VH, such that ⟨uv, wv′⟩ = ⟨uv′, wv⟩ = 0 whenever v and v′ are adjacent in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every vertex y ∈ VG, let Uy denote the subspace spanned by the vectors uv of the given orthogonal bi-representation associated with the vertices v of H whose tail is y, namely, Uy = span({uv | v = (x, y) for some x ∈ VG}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Similarly, let Wy denote the subspace spanned by the vectors wv of the given orthogonal bi- representation associated with the vertices v of H whose tail is y, namely, Wy = span({wv | v = (x, y) for some x ∈ VG}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that Uy and Wy are subspaces of Fn, hence the pair (Uy, Wy) is a vertex of S2(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Consider the function that maps every vertex y ∈ VG of G to the vertex (Uy, Wy) of S2(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We claim that this function forms a homomorphism from G to S2(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, let x, y ∈ VG be adjacent vertices in G, and consider the vectors u = u(x,y) and w = w(x,y) assigned by the 12 given orthogonal bi-representation to the vertex (x, y) of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By the definition of an orthogonal bi-representation, it holds that ⟨u, w⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since (x, y) is a vertex of H whose tail is y, it follows that u ∈ Uy and w ∈ Wy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Further, every vertex of H of the form (x′, x) for some x′ ∈ VG is adjacent in H to (x, y), hence it satisfies ⟨u(x′,x), w⟩ = ⟨u, w(x′,x)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since the subspaces Ux and Wx are spanned, respectively, by those vectors u(x′,x) and w(x′,x), we obtain that u is orthogonal to the subspace Wx and that w is orthogonal to the subspace Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It thus follows that the vectors u and w satisfy ⟨u, w⟩ ̸= 0, u ∈ Uy ∩ W⊥ x , and w ∈ Wy ∩ U⊥ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By symmetry, there also exist vectors u′, w′ ∈ Fn satisfying ⟨u′, w′⟩ ̸= 0, u′ ∈ Ux ∩ W⊥ y , and w′ ∈ Wx ∩ U⊥ y , hence the pairs (Ux, Wx) and (Uy, Wy) are adjacent vertices in S2(F, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We conclude that the above function is a homomorphism from G to S2(F, n), hence the chromatic numbers of these graphs satisfy χ(G) ≤ χ(S2(F, n)), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We derive the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let F be a finite field of size q, let G be a graph, and let H be the underlying graph of the digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, it holds that minrkF(H) ≥ � 1 2 · logq χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Put n = minrkF(H), and apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='12 to obtain that χ(G) ≤ χ(S2(F, n)) ≤ q2n2, where the second inequality holds because the number of vertices in S2(F, n) does not exceed q2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By rearranging, the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 The Chromatic Number of S2(R, n) We next consider the problem of determining the chromatic numbers of the graphs S2(R, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The following theorem shows that these graphs cannot be properly colored using a finite number of colors, in contrast to the graphs S1(R, n) addressed in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every integer n ≥ 3, it holds that χ(S2(R, n)) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Before proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='14, let us describe a significant difference between the behavior of ξR(G) and of minrkR(G) with respect to the chromatic number χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is not difficult to see that the chromatic number of a graph G is bounded from above by some function of ξR(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Indeed, given a k-dimensional orthogonal representation of a graph G over R, one can assign to each vertex the sign vector from {0, ±1}k of its vector, obtaining a proper coloring of G with at most 3k colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This implies that every graph G satisfies χ(G) ≤ 3ξR(G) (see also [33, Chapter 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' On the other hand, the chromatic number of a graph G cannot be bounded from above by any function of minrkR(G), as proved below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every integer m, there exists a graph G such that minrkR(G) ≤ 3 and yet χ(G) ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: For an integer n > 6, consider the ‘double shift graph’ Gn defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Its vertices are all the 3-subsets of [n], where two sets {x1, x2, x3} and {y1, y2, y3} with x1 < x2 < x3 and y1 < y2 < y3 are adjacent in Gn if either (x2, x3) = (y1, y2) or (x1, x2) = (y2, y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It was shown 13 in [13] that the graph Gn satisfies χ(Gn) = (1 + o(1)) · log log n (see also [14]), whereas its local chromatic number, a concept introduced by Erd¨os et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' [12], is known to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By an argument of Shanmugam, Dimakis, and Langberg [37, Theorem 1], this implies that minrkR(Gn) ≤ 3 (see also [2, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We are ready to derive Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='14: It clearly suffices to prove the assertion of the theorem for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let F denote the subgraph of S2(R, 3) induced by the pairs (U, W) of subspaces of R3 satisfying dim(U) = dim(W) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5, for every graph G with minrkR(G) ≤ 3, there exists a homomorphism from G to F and thus χ(G) ≤ χ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='15, the chromatic number of a graph G with minrkR(G) ≤ 3 can be arbitrarily large, hence χ(F) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since F is a subgraph of S2(R, 3), this yields that χ(S2(R, 3)) = ∞, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='3 Index Coding In this section, we study the optimal length of (not necessarily linear) index codes for the comple- ment of underlying graphs of line digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Recall Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We start by presenting an argument of Langberg and Sprintson [30, Theorem 4(a)] that relates the chromatic number of a graph to the length of an index code for its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In fact, we slightly modify their argument to obtain the improved bound stated below (with 2|Σ|k rather than |Σ||Σ|k in the statement of the result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let Σ be an alphabet of size at least 2, and let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If there exists an index code for G over Σ of length k, then χ(G) ≤ 2|Σ|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Assume without loss of generality that {0, 1} ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Put G = (V, E) and n = |V|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Suppose that there exists an index code for G over Σ of length k, and let E : Σn → Σk and gi : Σk+|NG(i)| → Σ for i ∈ V denote the corresponding encoding and decoding functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every vertex i ∈ V, we define a function hi : Σk → {0, 1} that determines for a given encoded message y ∈ Σk whether gi returns 0 on y when all the symbols of the side informa- tion of the ith receiver are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Formally speaking, for every y ∈ Σk, we define hi(y) = 0 if gi(y, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , 0) = 0, and hi(y) = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We claim that the assignment of the function hi to each vertex i ∈ V forms a proper coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, let i and j be adjacent vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let x ∈ Σn denote the vector with 1 in the ith entry and 0 everywhere else, and put y = E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By the correctness of the decoding functions, it follows that gi(y, x|NG(i)) = xi = 1 whereas gj(y, x|NG(j)) = xj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since i and j are adjacent in G, they are not adjacent in G, hence all the symbols in the side information x|NG(i) of i and in the side information x|NG(j) of j are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This implies that gi(y, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , 0) = 1 and gj(y, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , 0) = 0, and therefore hi(y) = 1 and hj(y) = 0, which yields that hi ̸= hj, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Finally, observe that the number of distinct functions hi : Σk → {0, 1} for i ∈ V does not exceed 2|Σ|k, implying that χ(G) ≤ 2|Σ|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We proceed by proving an analogue of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='16 for line digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 14 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let Σ be an alphabet of size at least 2, let G be a graph, and let H be the underlying graph of the digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If there exists an index code for H over Σ of length k, then χ(G) ≤ 2|Σ|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Assume without loss of generality that {0, 1} ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Put G = (VG, EG), H = (VH, EH), and n = |VH|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Recall that the vertices of H are the ordered pairs of adjacent vertices in G, hence n = 2 · |EG|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Suppose that there exists an index code for H over Σ of length k, and let E : Σn → Σk and g(u,v) : Σk+|NH(u,v)| → Σ for (u, v) ∈ VH denote the corresponding encoding and decoding functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every vertex v ∈ VG, we define a function hv : Σk → {0, 1} that determines for a given encoded message y ∈ Σk whether every function g(u,v) associated with a vertex (u, v) ∈ VH returns 0 on y when all the symbols in the side information of the receiver of the vertex (u, v) are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Formally speaking, for every y ∈ Σk, we define hv(y) = 0 if for every u ∈ VG with (u, v) ∈ VH, it holds that g(u,v)(y, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , 0) = 0, and hv(y) = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We claim that the assignment of the function hv to each vertex v ∈ VG forms a proper coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, let v1 and v2 be adjacent vertices in G, and notice that (v1, v2) is a vertex of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let x ∈ Σn denote the vector with 1 in the entry of (v1, v2) and 0 everywhere else, and put y = E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We first claim that hv1(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, consider any vertex (u, v1) ∈ VH, and notice that (u, v1) and (v1, v2) are adjacent in H and are thus not adjacent in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By the correctness of the decoding function g(u,v1), it follows that g(u,v1)(y, x|NH(u,v1)) = x(u,v1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since (u, v1) and (v1, v2) are not adjacent in H, all the symbols in the side information x|NH(u,v1) of the vertex (u, v1) are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We thus obtain that for every vertex u ∈ VG with (u, v1) ∈ VH, it holds that g(u,v1)(y, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' , 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By the definition of hv1, it follows that hv1(y) = 0, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We next claim that hv2(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To see this, observe that by the correctness of the decoding function g(v1,v2), it follows that g(v1,v2)(y, x|NH(v1,v2)) = x(v1,v2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It further holds that all the symbols in the side information x|NH(v1,v2) of the vertex (v1, v2) are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By the definition of hv2, it follows that hv2(y) = 1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We obtain that every two adjacent vertices v1 and v2 in G satisfy hv1 ̸= hv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Since the number of functions hv : Σk → {0, 1} for v ∈ VG does not exceed 2|Σ|k, it follows that χ(G) ≤ 2|Σ|k, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 4 Hardness Results In this section, we prove our hardness results for the orthogonality dimension and for minrank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We also suggest a potential avenue for proving hardness results for the general index coding problem over a constant-size alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The starting point of our hardness proofs is the following theorem of Wrochna and ˇZivn´y [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Recall that the function b : N → N is defined by b(n) = ( n ⌊n/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1 ([40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every integer k ≥ 4, it is NP-hard to decide whether a given graph G satisfies χ(G) ≤ k or χ(G) ≥ b(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Our hardness results for the orthogonality dimension and the minrank parameter over finite fields are given by the following theorem, which confirms Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 15 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' There exists a function f : N → N satisfying f(k) = (1 − o(1)) · � b(k) such that for every finite field F and for every sufficiently large integer k, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is NP-hard to decide whether a given graph G satisfies ξF(G) ≤ k or ξF(G) ≥ 1 √ log |F| · f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is NP-hard to decide whether a given graph G satisfies minrkF(G) ≤ k or minrkF(G) ≥ 1 √ 2·log |F| · f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: Fix a finite field F of size q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We start by proving the first item of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For an integer k ≥ 4, consider the problem of deciding whether a given graph G satisfies χ(G) ≤ b(k) or χ(G) ≥ b(b(k)), whose NP-hardness follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To obtain our hardness result on the orthogonality dimension over F, we reduce from this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Consider the reduction that given an input graph G produces and outputs the underlying graph H of the digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This reduction can clearly be implemented in polynomial time (in fact, in logarithmic space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To prove the correctness of the reduction, we analyze the orthogonality dimension of H over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If G is a YES instance, that is, χ(G) ≤ b(k), then by combining Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6 with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2, it follows that ξF(H) ≤ χ(H) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If G is a NO instance, that is, χ(G) ≥ b(b(k)), then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5, it follows that ξF(H) ≥ � logq χ(G) ≥ � logq b(b(k)) = 1−o(1) √ log q · � b(k), where the o(1) term tends to 0 when k tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Note that we have used here the fact that b(n) = Θ(2n/√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By letting k be any sufficiently large integer, the proof of the first item of the theorem is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' The proof of the second item of the theorem is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To avoid repetitions, we briefly mention the needed changes in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' First, to obtain a hardness result for the minrank parameter, the reduction has to output the complement H of the graph H rather than H itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Second, in the analysis of the NO instances, one has to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='13 instead of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5 to obtain that minrkF(H) ≥ � 1 2 · logq χ(G) ≥ � 1 2 · logq b(b(k)) = 1−o(1) √ 2·log q · � b(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' As an immediate corollary of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For every finite field F, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is NP-hard to approximate ξF(G) for a given graph G to within any constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is NP-hard to approximate minrkF(G) for a given graph G to within any constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We next prove a hardness result for the orthogonality dimension over the reals, confirming Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' There exists a function f : N → N satisfying f(k) = Θ( � b(k)/k) such that for every sufficiently large integer k, it is NP-hard to decide whether a given graph G satisfies ξR(G) ≤ k or ξR(G) ≥ f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Proof: As in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2, for an integer k ≥ 4, we reduce from the problem of deciding whether a given graph G satisfies χ(G) ≤ b(k) or χ(G) ≥ b(b(k)), whose NP-hardness follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Consider the polynomial-time reduction that given an input graph G produces and outputs the underlying graph H of the digraph δG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' To prove the correctness of the reduction, we analyze the orthogonality dimension of H over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If G is a YES instance, that is, χ(G) ≤ b(k), then by combining Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6 with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2, it follows that ξR(H) ≤ χ(H) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' If G is a NO instance, that is, χ(G) ≥ b(b(k)), then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7 combined with the fact that b(n) = Θ(2n/√n), it follows that ξR(H) ≥ c · � log b(b(k)) log log b(b(k)) = Θ �� b(k) k � , where c is an absolute positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' As an immediate corollary of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='4, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' It is NP-hard to approximate ξR(G) for a given graph G to within any constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We end this section with a statement that might be useful for proving NP-hardness results for the general index coding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Consider the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For an alphabet Σ and for two integers k1 < k2, let Index-CodingΣ(k1, k2) denote the problem of deciding whether the minimal length of an index code for a given graph G over Σ is at most k1 or at least k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Let Σ be an alphabet of size at least 2, and let k1, k2 be two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, there exists a polynomial-time reduction from the problem of deciding whether a given graph G satisfies χ(G) ≤ b(k1) or χ(G) ≥ k2 to Index-CodingΣ(k1, log|Σ| log k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' 17 Proof: Consider the polynomial-time reduction that given an input graph G produces the under- lying graph H of the digraph δG and outputs its complement H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' For correctness, suppose first that G is a YES instance, that is, χ(G) ≤ b(k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Then, by combining Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='6 with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='2, it follows that minrkF2(H) ≤ χ(H) ≤ k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='16, it further follows that there exists a linear index code for H over F2 of length k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' In particular, using |Σ| ≥ 2, there exists an index code for H over the alphabet Σ of length k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Suppose next that G is a NO instance, that is, χ(G) ≥ k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='17, it follows that the length of any index code for H over Σ is at least log|Σ| log k2, so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content='7 implies that in order to prove the NP-hardness of the general index coding prob- lem over some finite alphabet Σ of size at least 2, it suffices to prove for some integer k that it is NP-hard to decide whether a given graph G satisfies χ(G) ≤ b(k) or χ(G) > 2|Σ|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Acknowledgements We thank the anonymous reviewers for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQf1PmO/content/2301.00732v1.pdf'} +page_content=' Alon.' metadata={'source': 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