diff --git "a/ANE1T4oBgHgl3EQfVQQy/content/tmp_files/load_file.txt" "b/ANE1T4oBgHgl3EQfVQQy/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/ANE1T4oBgHgl3EQfVQQy/content/tmp_files/load_file.txt" @@ -0,0 +1,971 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf,len=970 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='03099v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='AI] 8 Jan 2023 Fully Dynamic Online Selection through Online Contention Resolution Schemes Vashist Avadhanula*, Andrea Celli1, Riccardo Colini-Baldeschi2, Stefano Leonardi3, Matteo Russo3 1Department of Computing Sciences, Bocconi University, Milan, Italy 2 Core Data Science, Meta, London, UK 3Department of Computer, Control and Management Engineering, Sapienza University, Rome, Italy vas1089@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='com, andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='celli2@unibocconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='it, rickuz@fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='com, leonardi@diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='it, mrusso@diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='it Abstract We study fully dynamic online selection problems in an ad- versarial/stochastic setting that includes Bayesian online se- lection, prophet inequalities, posted price mechanisms, and stochastic probing problems subject to combinatorial con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the classical “incremental” version of the problem, selected elements remain active until the end of the input se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' On the other hand, in the fully dynamic version of the problem, elements stay active for a limited time interval, and then leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This models, for example, the online matching of tasks to workers with task/worker-dependent working times, and sequential posted pricing of perishable goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A success- ful approach to online selection problems in the adversarial setting is given by the notion of Online Contention Resolution Scheme (OCRS), that uses a priori information to formulate a linear relaxation of the underlying optimization problem, whose optimal fractional solution is rounded online for any adversarial order of the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Our main contribu- tion is providing a general method for constructing an OCRS for fully dynamic online selection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, we show how to employ such OCRS to construct no-regret algorithms in a partial information model with semi-bandit feedback and adversarial inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 1 Introduction Consider the case where a financial service provider receives multiple operations every hour/day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' These operations might be malicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The provider needs to assign them to human reviewers for inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The time required by each reviewer to file a reviewing task and the reward (weight) that is ob- tained with the review follow some distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The distri- butions can be estimated from historical data, as they depend on the type of transaction that needs to be examined and on the expertise of the employed reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' To efficiently solve the problem, the platform needs to compute a matching be- tween tasks and reviewers based on the a priori information that is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' However, the time needed for a specific re- view, and the realized reward (weight), is often known only after the task/reviewer matching is decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A multitude of variations to this setting are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For instance, if a cost is associated with each reviewing task, the total cost for the reviewing process might be bounded by a budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Moreover, there might be various kinds of restric- tions on the subset of reviewers that are assigned at each Research performed while the author was working at Meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Finally, the objective function might not only be the sum of the rewards (weights) we observe, if, for example, the decision maker has a utility function with “diminishing return” property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' To model the general class of sequential decision prob- lems described above, we introduce fully dynamic online se- lection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This model generalizes online selection problems (Chekuri, Vondr´ak, and Zenklusen 2011), where elements arrive online in an adversarial order and algorithms can use a priori information to maximize the weight of the selected subset of elements, subject to combinatorial con- straints (such as matroid, matching, or knapsack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the classical version of the problem (Chekuri, Vondr´ak, and Zenklusen 2011), once an element is selected, it will af- fect the combinatorial constraints throughout the entire input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This is in sharp contrast with the fully dynamic version, where an element will affect the combinatorial con- straint only for a limited time interval, which we name ac- tivity time of the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For example, a new task can be matched to a reviewer as soon as she is done with previ- ously assigned tasks, or an agent can buy a new good as soon as the previously bought goods are perished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A large class of Bayesian online selection (Kleinberg and Weinberg 2012), prophet inequality (Hajiaghayi, Kleinberg, and Sand- holm 2007), posted price mechanism (Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2010), and stochastic probing (Gupta and Nagarajan 2013) prob- lems that have been studied in the classical version of on- line selection can therefore be extended to the fully dynamic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Note that in the dynamic algorithms literature, fully dynamic algorithms are algorithms that deal with both adver- sarial insertions and deletions (Demetrescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We could also interpret our model in a similar sense since ele- ments arrive online (are inserted) according to an adversarial order, and cease to exist (are deleted) according to adversar- ially established activity times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A successful approach to online selection problems is based on Online Contention Resolution Schemes (OCRSs) (Feldman, Svensson, and Zenklusen 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' OCRSs use a priori information on the values of the elements to formu- late a linear relaxation whose optimal fractional solution up- per bounds the performance of the integral offline optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, an online rounding procedure is used to produce a so- lution whose value is as close as possible to the fractional re- laxation solution’s value, for any adversarial order of the in- put sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The OCRS approach allows to obtain good ap- proximations of the expected optimal solution for linear and submodular objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The existence of OCRSs for fully dynamic online selection problems is therefore a natu- ral research question that we address in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The OCRS approach is based on the availability of a pri- ori information on weights and activity times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' However, in real world scenarios, these might be missing or might be expensive to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Therefore, in the second part of our work, we study the fully dynamic online selection problem with partial information, where the main research question is whether the OCRS approach is still viable if a priori in- formation on the weights is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In order to answer this question, we study a repeated version of the fully dynamic online selection problem, in which at each stage weights are unknown to the decision maker (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', no a priori informa- tion on weights is available) and chosen adversarially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The goal in this setting is the design of an online algorithm with performances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', cumulative sum of weights of selected elements) close to that of the best static selection strategy in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Our Contributions First, we introduce the fully dynamic online selection prob- lem, in which elements arrive following an adversarial or- dering, and revealed one-by-one their weights and activ- ity times at the time of arrival (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', prophet model), or after the element has been selected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', probing model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Our model describes temporal packing constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', downward-closed), where elements are active only within their activity time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The objective is to maximize the weight of the selected set of elements subject to temporal packing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We provide two black-box reductions for adapting classical OCRS for online (non-dynamic) se- lection problems to the fully dynamic setting under full and partial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Blackbox reduction 1: from OCRS to temporal OCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Starting from a (b, c)-selectable greedy OCRS in the clas- sical setting, we use it as a subroutine to build a (b, c)- selectable greedy OCRS in the more general temporal setting (see Algorithm 1 and Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This means that competi- tive ratio guarantees in one setting determine the same guar- antees in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Such a reduction implies the existence of algorithms with constant competitive ratio for online opti- mization problems with linear or submodular objective func- tions subject to matroid, matching, and knapsack constraints, for which we give explicit constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We also extend the framework to elements arriving in batches, which can have correlated weights or activity times within the batch, as de- scribed in the appendix of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Blackbox reduction 2: from temporal OCRS to no-α- regret algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Following the recent work by Gergatsouli and Tzamos (2022) in the context of Pandora’s box prob- lems, we define the following extension of the problem to the partial-information setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For each of the T stages, the al- gorithm is given in input a new instance of the fully dynamic online selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Activity times are fixed before- hand and known to the algorithm, while weights are chosen by an adversary, and revealed only after the selection at the current stage has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In such setting, we show that an α-competitive temporal OCRS can be exploited in the adversarial partial-information version of the problem, in order to build no-α-regret algorithms with polynomial per- iteration running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Regret is measured with respect to the cumulative weights collected by the best fixed selection pol- icy in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We study three different settings: in the first setting, we study the full-feedback model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', the algorithm observes the entire utility function at the end of each stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, we focus on the semi-bandit-feedback model, in which the algorithm only receives information on the weights of the elements it selects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In such setting, we provide a no-α-regret framework with ˜O(T 1/2) upper bound on cumulative regret in the case in which we have a “white-box” OCRS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', we know the exact procedure run within the OCRS, and we are able to simulate it ex-post).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Moreover, we also provide a no- α-regret algorithm with ˜O(T 2/3) regret upper bound for the case in which we only have oracle access to the OCRS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', the OCRS is treated as a black-box, and the algorithm does not require knowledge about its internal procedures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Related Work In the first part of the paper, we deal with a setting where the algorithm has complete information over the input but is unaware of the order in which elements arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In this con- text, Contention resolution schemes (CRS) were introduced by Chekuri, Vondr´ak, and Zenklusen (2011) as a powerful rounding technique in the context of submodular maximiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The CRS framework was extended to online contention resolution schemes (OCRS) for online selection problems by Feldman, Svensson, and Zenklusen (2016), who provided constant competitive OCRSs for different problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' in- tersections of matroids, matchings, and prophet inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We generalize the OCRS framework to a setting where ele- ments are timed and cease to exist right after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the second part, we lift the complete knowledge as- sumption and work in an adversarial bandit setting, where at each stage the entire set of elements arrives, and we seek to select the “best” feasible subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This is similar to the problem of combinatorial bandits (Cesa-Bianchi and Lugosi 2012), but unlike it, we aim to deal with combinatorial se- lection of timed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In this respect, blocking bandits (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2019) model situations where played arms are blocked for a specific number of stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Despite their con- textual (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2021), combinatorial (Atsidakou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2021), and adversarial (Bishop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2020) extensions, re- cent work on blocking bandits only addresses specific cases of the fully dynamic online selection problem (Dickerson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2018), which we solve in entire generality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' adver- sarially and for all packing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Our problem is also related to sleeping bandits (Klein- berg, Niculescu-Mizil, and Sharma 2010), in that the adver- sary decides which actions the algorithm can perform at each stage t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Nonetheless, a sleeping bandit adversary has to com- municate all available actions to the algorithm before a stage starts, whereas our adversary sets arbitrary activity times for each element, choosing in what order elements arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2 Preliminaries Given a finite set X ⊆ Rn and Y ⊆ 2X , let 1Y ∈ {0, 1}|X| be the characteristic vector of set X, and co X be the convex hull of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We denote vectors by bold fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given vector x, we denote by xi its i-th component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' , n}, with n ∈ N>0, is compactly denoted as [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a set X and a scalar α ∈ R, let αX := {αx : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Finally, given a discrete set X, we denote by ∆X the |X|-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We start by introducing a general selection problem in the standard (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', non-dynamic) case as studied by Kleinberg and Weinberg (2012) in the context of prophet inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let E be the ground set and let m := |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Each element e ∈ E is characterized by a collection of parameters ze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In gen- eral, ze is a random variable drawn according to an element- specific distribution ζe, supported over the joint set of pos- sible parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the standard (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', non-dynamic) setting, ze just encodes the weight associated to element e, that is ze = (we), for some we ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='1 In such case distributions ζe are supported over [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Random variables {ze : e ∈ E} are independent, and ze is distributed according to ζe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An in- put sequence is an ordered sequence of elements and weights such that every element in E occurs exactly once in the se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The order is specified by an arrival time se for each element e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Arrival times are such that se ��� [m] for all e ∈ E, and for two distinct e, e′ we have se ̸= se′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The order of arrival of the elements is a priori unknown to the algorithm, and can be selected by an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the standard full- information setting the distributions ζe can be chosen by an adversary, but they are known to the algorithm a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We consider problems characterized by a family of packing con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Definition 1 (Packing Constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A family of constraints F = (E, I), for ground set E and independence family I ⊆ 2E, is said to be packing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', downward-closed) if, taken A ∈ I, and B ⊆ A, then B ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Elements of I are called independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Such family of constraints is closed under intersection, and encompasses matroid, knapsack, and matching constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Fractional LP formulation Even in the offline setting, in which the ordering of the input sequence (se)e∈E is known beforehand, determining an independent set of maximum cumulative weight may be NP-hard in the worst-case (Feige 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, we consider the relaxation of the problem in which we look for an optimal fractional solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The value of such solution is an upper bound to the value of the true offline optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Therefore, any algorithm guar- anteeing a constant approximation to the offline fractional optimum immediately yields the same guarantees with re- spect to the offline optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a family of packing con- straints F = (E, I), in order to formulate the problem of computing the best fractional solution as a linear program- ming problem (LP) we introduce the notion of packing con- straint polytope PF ⊆ [0, 1]m which is such that PF := co ({1S : S ∈ I}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a non-negative submodular func- tion f : [0, 1]m → R≥0, and a family of packing constraints 1This is for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the dynamic case ze will contain other parameters in addition to weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' F, an optimal fractional solution can be computed via the LP maxx∈PF f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' If the goal is maximizing the cumula- tive sum of weights, the objective of the optimization prob- lem is ⟨x, w⟩, where w := (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' , wm) ∈ [0, 1]m is a vector specifying the weight of each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' If we assume access to a polynomial-time separation oracle for PF such LP yields an optimal fractional solution in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Online selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the online version of the prob- lem, given a family of packing constraints F, the goal is se- lecting an independent set whose cumulative weight is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In such setting, the elements reveal one by one their realized ze, following a fixed prespecified order unknown to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Each time an element reveals ze, the algorithm has to choose whether to select it or discard it, before the next element is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Such decision is irrevo- cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Computing the exact optimal solution to such online selection problems is intractable in general (Feige 1998), and the goal is usually to design approximation algorithms with good competitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='2 In the remainder of the section we describe one well-known framework for such objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Online contention resolution schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Contention resolu- tion schemes were originally proposed by Chekuri, Vondr´ak, and Zenklusen (2011) in the context of submodular function maximization, and later extended to online selection prob- lems by Feldman, Svensson, and Zenklusen (2016) under the name of online contention resolution schemes (OCRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a fractional solution x ∈ PF, an OCRS is an online rounding procedure yielding an independent set in I guar- anteeing a value close to that of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let R(x) be a random set containing each element e independently and with prob- ability xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The set R(x) may not be feasible according to constraints F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An OCRS essentially provides a procedure to construct a good feasible approximation by starting from the random set R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Formally, Definition 2 (OCRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a point x ∈ PF and the set of elements R(x), elements e ∈ E reveal one by one whether they belong to R(x) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An OCRS chooses irrevocably whether to select an element in R(x) before the next element is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An OCRS for PF is an online algorithm that selects S ⊆ R(x) such that 1S ∈ PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We will focus on greedy OCRS, which were defined by Feld- man, Svensson, and Zenklusen (2016) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Definition 3 (Greedy OCRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let PF ⊆ [0, 1]m be the fea- sibility polytope for constraint family F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An OCRS π for PF is called a greedy OCRS if, for every ex-ante feasible solu- tion x ∈ PF, it defines a packing subfamily of feasible sets Fπ,x ⊆ F, and an element e is selected upon arrival if, to- gether with the set of already selected elements, the resulting set is in Fπ,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A greedy OCRS is randomized if, given x, the choice of Fπ,x is randomized, and deterministic otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For b, c ∈ [0, 1], we say that a greedy OCRS π is (b, c)-selectable if, for each e ∈ E, and given x ∈ bPF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', belonging to a 2The competitive ratio is computed as the worst-case ratio be- tween the value of the solution found by the algorithm and the value of an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' down-scaled version of PF), Prπ,R(x) [S ∪ {e} ∈ Fπ,x ∀S ⊆ R(x), S ∈ Fπ,x] ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Intuitively, this means that, with probability at least c, the random set R(x) is such that an element e is selected no matter what other elements I of R(x) have been selected so far, as long as I ∈ Fπ,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This guarantees that an ele- ment is selected with probability at least c against any ad- versary, which implies a bc competitive ratio with respect to the offline optimum (see Appendix A for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Now, we provide an example due to Feldman, Svensson, and Zenklusen (2016) of a feasibility constraint family where OCRSs guarantee a constant competitive ratio against the offline optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We will build on this example throughout the paper in order to provide intuition for the main concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Example 1 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='7 in (Feldman, Svensson, and Zen- klusen 2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a graph G = (V, E), with |E| = m edges, we consider a matching feasibility polytope PF = � x ∈ [0, 1]m : � e∈δ(u) xe ≤ 1, ∀u ∈ V � , where δ(u) de- notes the set of all adjacent edges to u ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given b ∈ [0, 1], the OCRS takes as input x ∈ bPF, and samples each edge e with probability xe to build R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, it selects each edge e ∈ R(x), upon its arrival, with probability (1 − e−xe)/xe only if it is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, the probability to select any edge e = (u, v) (conditioned on being sampled) is 1 − e−xe xe � e′∈δ(u)∪δ(v)\\{e} e−xe′ = 1 − e−xe xe e− � e′∈δ(u)∪δ(v)\\{e} xe′ ≥ 1 − e−xe xe e−2b ≥ e−2b, where the inequality follows from xe ∈ bPF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', � e′∈δ(u)\\{e} xe′ ≤ b−xe, and similarly for δ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Note that in order to obtain an unconditional probability, we need to multiply the above by a factor xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We remark that this example resembles closely our in- troductory motivating application, where financial transac- tions need to be assigned to reviewers upon their arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Moreover, Feldman, Svensson, and Zenklusen (2016) give explicit constructions of (b, c)-selectable greedy OCRSs for knapsack, matching, matroidal constraints, and their inter- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We include a discussion of their feasibility poly- topes in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Ezra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' (2020) generalize the above online selection procedure to a setting where elements arrive in batches rather than one at a time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' we provide a discussion of such setting in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 3 Fully Dynamic Online Selection The fully dynamic online selection problem is characterized by the definition of temporal packing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We gen- eralize the online selection model (Section 2) by introduc- ing an activity time de ∈ [m] for each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Element e arrives at time se and, if it is selected by the algorithm, it re- mains active up to time se + de and “blocks” other elements from being selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Elements arriving after that time can be selected by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In this setting, each element e ∈ E is characterized by a tuple of attributes ze := (we, de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let Fd := (E, Id) be the family of temporal packing fea- sibility constraints where elements block other elements in the same independent set according to activity time vector d = (de)e∈E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The goal of fully dynamic online selection is selecting an independent set in Id whose cumulative weight is as large as possible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', as close as possible to the offline optimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We can naturally extend the expression for pack- ing polytopes in the standard setting to the temporal one for every feasibility constraint family, by exploiting the follow- ing notion of active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Definition 4 (Active Elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For element e ∈ E and given {ze}e∈E, we denote the set of active elements as Ee := {e′ ∈ E : se′ ≤ se ≤ se′ + de′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='3 In this setting, we don’t need to select an independent set S ∈ F, but, in a less restrictive way, we only require that for each incoming element we select a feasible subset of the set of active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Definition 5 (Temporal packing constraint polytope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given F = (E, I), a temporal packing constraint polytope Pd F ⊆ [0, 1]m is such that Pd F := co ({1S : S ∩ Ee ∈ I, ∀e ∈ E}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For a fixed element e, the temporal polytope is the convex hull of the collection containing all the sets such that S ∩ Ee is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This needs to be true for all e ∈ E, meaning that we can rewrite the polytope and the feasibility set as Pd F = co �� e∈E {1S : S ∩ Ee ∈ F} � , and Id = � e∈E {S : S ∩ Ee ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Moreover, when d and d′ differ for at least one element e, that is de < d′ e, then Ee ⊆ E′ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, Pd F ⊇ Pd′ F , Id ⊇ Id′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We now extend Example 1 to account for activity times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In Appendix B we also work out the reduction from stan- dard to temporal packing constraints for a number of exam- ples, including rank-1 matroids (single-choice), knapsack, and general matroid constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We consider the temporal extension of the matching polytope presented in Example 1, that is Pd F = \uf8f1 \uf8f2 \uf8f3y ∈ [0, 1]m : � e∈δ(u)∩Ee xe ≤ 1, ∀u ∈ V, ∀e ∈ E \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let us use the same OCRS as in the previous example, but where “feasibility” only concerns the subset of active edges in δ(u) ∪ δ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The probability to select an edge e = (u, v) is 1 − e−xe xe � e′∈δ(u)∪δ(v)∩Ee\\{e} e−xe′ ≥ 1 − e−xe xe e−2b ≥ e−2b, which is obtained in a similar way to Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The above example suggests to look for a general reduc- tion that maps an OCRS for the standard setting, to an OCRS for the temporal setting, while achieving at least the same competitive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 3Note that, since for distinct elements e, e′, we have se′ ̸= se, we can equivalently define the set of active elements as Ee := {e′ ∈ E : se′ < se ≤ se′ + de′} ∪ {e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Algorithm 1: Greedy OCRS Black-box Reduction Input: Feasibility families F and Fd, polytopes PF and Pd F, OCRS π for F, a point x ∈ bPd F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Initialize Sd ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Sample R(x) such that Pr [e ∈ R(x)] = xe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' for e ∈ E do Upon arrival of element e, compute the set of currently active elements Ee;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' if (Sd ∩ Ee) ∪ {e} ∈ Fπ,y then Execute the original greedy OCRS π(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Update Sd accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' else Discard element e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' return set Sd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 4 OCRS for Fully Dynamic Online Selection The first black-box reduction which we provide consists in showing that a (b, c)-selectable greedy OCRS for stan- dard packing constraints implies the existence of a (b, c)- selectable greedy OCRS for temporal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In partic- ular, we show that the original greedy OCRS working for x ∈ bPF can be used to construct another greedy OCRS for y ∈ bPd F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' To this end, Algorithm 1 provides a way of exploiting the original OCRS π in order to manage temporal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For each element e, and given the induced sub- family of packing feasible sets Fπ,y, the algorithm checks whether the set of previously selected elements Sd which are still active in time, together with the new element e, is feasible with respect to Fπ,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' If that is the case, the algo- rithm calls the OCRS π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, if the OCRS π for input y decided to select the current element e, the algorithm adds it to Sd, otherwise the set remains unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We remark that such a procedure is agnostic to whether the original greedy OCRS is deterministic or randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We observe that, due to a larger feasibility constraint family, the number of in- dependent sets have increased with respect to the standard setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' However, we show that this does not constitute a problem, and an equivalence between the two settings can be established through the use of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The follow- ing result shows that Algorithm 1 yields a (b, c)-selectable greedy OCRS for temporal packing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let F, Fd be the standard and temporal pack- ing constraint families, respectively, and let their corre- sponding polytopes be PF and Pd F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let x ∈ bPF and y ∈ bPd F, and consider a (b, c)-selectable greedy OCRS π for Fπ,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, Algorithm 1 equippend with π is a (b, c)- selectable greedy OCRS for Fd π,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let us denote by ˆπ the procedure described in Algo- rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' First, we show that ˆπ is a greedy OCRS for Fd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Greedyness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' It is clear from the setting and the construc- tion that elements arrive one at a time, and that ˆπ irrevoca- bly selects an incoming element only if it is feasible, and before seeing the next element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Indeed, in the if statement of Algorithm 1, we check that the active subset of the el- ements selected so far, together with the new arriving ele- ment e, is feasible against the subfamily Fπ,x ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Con- straint subfamily Fπ,x is induced by the original OCRS π, and point x belongs to the polytope bPd F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Note that we do not necessarily add element e to the running set Sd, even though feasible, but act as the original greedy OCRS would have acted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' All that is left to be shown is that such a procedure defines a subfamily of feasibility constraints Fd π,x ⊆ Fd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' By construction, on the arrival of each element e, we guarantee that Sd is a set such that its subset of active elements is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This means that Sd ∩ Ee ∈ Fπ,x ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, Sd ∈ Fd π,x := � e∈E {S : S ∩ Ee ∈ Fπ,x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Finally, Fπ,x ⊆ F implies that Fd π,x ⊆ Fd, which shows that ˆπ is greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' With the above, we can now turn to demonstrate (b, c)-selectability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Selectability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Upon arrival of element e ∈ E, let us con- sider S and Sd to be the sets of elements already selected by π and ˆπ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' By the way in which the con- straint families are defined, and by construction of ˆπ, we can observe that, given x ∈ bPd F and y ∈ bPF, for all S ⊆ R(y) such that S ∪{e} ∈ Fπ,y, there always exists a set Sd ⊆ R(x) such that (Sd ∩Ee)∪{e} ∈ Fπ,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This es- tablishes an injection between the selected set under stan- dard constraints, and its counterpart under temporal con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We observe that, for all e ∈ E and x ∈ bPd F, Pr � Sd ∪ {e} ∈ Fd π,x ∀Sd ⊆ R(x), Sd ∈ Fd π,x � = Pr � (Sd ∩ Ee) ∪ {e} ∈ Fπ,x ∀Sd ⊆ R(x), Sd ∩ Ee ∈ Fd π,x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Hence, since for greedy OCRS π and y ∈ bPF, we have that Pr [S ∪ {e} ∈ Fπ,y ∀S ⊆ R(y), S ∈ Fπ,y] ≥ c, we can conclude by the injection above that Pr � (Sd ∩ Ee) ∪ {e} ∈ Fπ,x ∀Sd ⊆ R(x), Sd ∩ Ee ∈ Fπ,x � ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We remark that the above reduction is agnostic to the weight scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', we need not assume that we ∈ [0, 1] for all e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In order to further motivate the significance of Algorithm 1 and Theorem 1, in the Appendix we explic- itly reduce the standard setting to the fully dynamic one for single-choice, and provide a general recipe for all packing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 5 Fully Dynamic Online Selection under Partial Information In this section, we study the case in which the decision- maker has to act under partial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In particular, we focus on the following online sequential extension of the full-information problem: at each stage t ∈ [T ], a de- cision maker faces a new instance of the fully dynamic online selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An unknown vector of weights wt ∈ [0, 1]|E| is chosen by an adversary at each stage t, while feasibility set Fd is known and fixed across all T stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This setting is analogous to the one recently stud- ied by Gergatsouli and Tzamos (2022) in the context of Pandora’s box problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A crucial difference with the on- line selection problem with full-information studied in Sec- tion 4 is that, at each step t, the decision maker has to de- cide whether to select or discard an element before observ- ing its weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In particular, at each t, the decision maker takes an action at := 1Sd t , where Sd t ∈ Fd is the fea- sible set selected at stage t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The choice of at is made be- fore observing wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The objective of maximizing the cumu- lative sum of weights is encoded in the reward function f : [0, 1]2m ∋ (a, w) �→ ⟨a, w⟩ ∈ [0, 1], which is the reward obtained by playing a with weights w = (we)e∈E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 4 In this setting, we can think of Fd as the set of super-arms in a combinatorial online optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Our goal is designing online algorithms which have a performance close to that of the best fixed super-arm in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='5 In the analy- sis, as it is customary when the online optimization problem has an NP-hard offline counterpart, we resort to the notion of α-regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In particular, given a set of feasible actions X, we define an algorithm’s α-regret up to time T as Regretα(T ) := α max x∈X � T � t=1 f(x, wt) � −E � T � t=1 f(xt, wt) � , where α ∈ (0, 1] and xt is the strategy output by the online algorithm at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We say that an algorithm has the no-α- regret property if Regretα(T )/T → 0 for T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The main result of the section is providing a black-box re- duction that yields a no-α-regret algorithm for any fully dy- namic online selection problem admitting a temporal OCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We provide no-α-regret frameworks for three scenarios: full-feedback model: after selecting at the decision-maker observes the exact reward function f(·, wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' semi-bandit feedback with white-box OCRS: after taking a decision at time t, the algorithm observes wt,e for each el- ement e ∈ Sd t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', each element selected at t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Moreover, the decision-maker has exact knowledge of the procedure employed by the OCRS, which can be easily simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' semi-bandit feedback with oracle access to the OCRS: the decision maker has semi-bandit feedback and the OCRS is given as a black-box which can be queried once per step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Full-feedback Setting In this setting, after selecting at, the decision-maker gets to observe the reward function f(·, wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In order to achieve performance close to that of the best fixed super-harm in hindsight the idea is to employ the α-competitive OCRS de- signed in Section 4 by feeding it with a fractional solution 4The analysis can be easily extended to arbitrary functions lin- ear in both terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 5As we argue in Appendix D it is not possible to be competitive with respect to more powerful benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Algorithm 2: FULL-FEEDBACK ALGORITHM Input: T , Fd, temporal OCRS ˆπ, subroutine RM Initialize RM for strategy space Pd F for t ∈ [T ] do xt ← RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='RECOMMEND() at ← execute OCRS ˆπ with input xt Play at, and subsequently observe f(·, wt) RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='UPDATE(f(·, wt)) xt computed by considering the weights selected by the ad- versary up to time t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='6 Let us assume to have at our disposal a no-α-regret algo- rithm for decision space Pd F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We denote such regret min- imizer as RM, and we assume it offers two basic opera- tions: i) RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='RECOMMEND() returns a vector in Pd F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' ii) RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='UPDATE(f(·, w)) updates the internal state of the re- gret minimizer using feedback received by the environment in the form of a reward function f(·, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Notice that the availability of such component is not enough to solve our problem since at each t we can only play a super-arm at ∈ {0, 1}m feasible for Fd, and not the strategy xt ∈ Pd F ⊆ [0, 1]m returned by RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The decision-maker can exploit the subroutine RM together with a temporal greedy OCRS ˆπ by following Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We can show that, if the algorithm employs a regret minimizer for Pd F with a sublinear cumu- lative regret upper bound of RT , the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a regret minimizer RM for decision space Pd F with cumulative regret upper bound RT , and an α-competitive temporal greedy OCRS, Algorithm 2 provides α max S∈Id T � t=1 f(1S, wt) − E � T � t=1 f(at, wt) � ≤ RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Since we are assuming the existence of a polynomial- time separation oracle for the set Pd F, then the LP arg maxx∈Pd F f(x, w) can be solved in polynomial time for any w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Therefore, we can instantiate a regret minimizer for Pd F by using, for example, follow-the-regularised-leader which yields RT ≤ ˜O(m √ T) (Orabona 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Semi-Bandit Feedback with White-Box OCRS In this setting, given a temporal OCRS ˆπ, it is enough to show that we can compute the probability that a certain super-arm a is selected by ˆπ given a certain order of ar- rivals at stage t and a vector of weights w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' If that is the case, we can build a no-α-regret algorithm with regret upper bound of ˜O(m √ T) by employing Algorithm 2 and by in- stantiating the regret minimizer RM as the online stochastic mirror descent (OSMD) framework by Audibert, Bubeck, and Lugosi (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We observe that the regret bound ob- tained is this way is tight in the semi-bandit setting (Audib- ert, Bubeck, and Lugosi 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let qt(e) be the probability 6We remark that a (b, c)-selectable OCRS yields a bc competi- tive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In the following, we let α := bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Algorithm 3: SEMI-BANDIT-FEEDBACK ALGO- RITHM WITH ORACLE ACCESS TO OCRS Input: T , Fd, temporal OCRS ˆπ, full-feedback algorithm RM for decision space Pd F Let Z be initialized as in Theorem 3, and initialize RM appropriately for τ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' , Z do Iτ ← � (τ − 1) T Z + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' , τ T Z � Choose a random permutation p : [m] → E, and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' , tm stages at random from Iτ xτ ← RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='RECOMMEND() for t = (τ − 1) T Z + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' , τ T Z do if t = tj for some j ∈ [m] then xt ← 1Sd for a feasible set Sd containing p(j) elsext ← xτ Play at obtained from the OCRS ˆπ executed with fractional solution xt Compute estimators ˜fτ(e) of fτ(e) := 1 |Iτ | � t∈Iτ f(1e, wt) for each e ∈ E RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='UPDATE � ˜fτ(·) � with which our algorithm selects element e at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, we can equip OSMD with the following unbiased estimator of the vector of weights: ˆwt,e := wt,eat,e/qt(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 7 In order to compute qt(·) we need to have observed the order of arrival at stage t, the weights corresponding to super-arm at, and we need to be able to compute the probability with which the OCRS selected e at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This the reason for which we talk about “white-box” OCRS, as we need to simulate ex post the procedure followed by the OCRS in order to compute qt(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' When we know the procedure followed by the OCRS, we can always compute qt(e) for any element e selected at stage t, since at the end of stage t we know the order of arrival, weights for selected elements, and the initial frac- tional solution xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We provide further intuition as for how to compute such probabilities through the running example of matching constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Consider Algorithm 2 initialized with the OCRS of Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given stage t, we can safely limit our atten- tion to selected edges (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', elements e such that at,e = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Indeed, all other edges will either be unfeasible (which im- plies that the probability of selecting them is 0), or they were not selected despite being feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Consider an arbitrary element e among those selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Conditioned on the past choices up to element e, we know that e ∈ at will be fea- sible with certainty, and thus the (unconditional) probability it is selected is simply qt(e) = 1 − e−yt,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Semi-Bandit Feedback and Oracle Access to OCRS As in the previous case, at each stage t the decision maker can only observe the weights associated to each edge se- 7We observe that ˆwt,e is equal to 0 when e has not been selected at stage t because, in that case, at,e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' lected by at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Therefore, they have no counterfactual infor- mation on their reward had they selected a different feasi- ble set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' On top of that, we assume that the OCRS is given as a black-box, and therefore we cannot compute ex post the probabilities qt(e) for selected elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' However, we show that it is possible to tackle this setting by exploiting a reduction from the semi-bandit feedback setting to the full- information feedback one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In doing so, we follow the ap- proach first proposed by Awerbuch and Kleinberg (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The idea is to split the time horizon T into a given num- ber of equally-sized blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Each block allows the decision maker to simulate a single stage of the full information set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We denote the number of blocks by Z, and each block τ ∈ [Z] is composed by a sequence of consecutive stages Iτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Algorithm 3 describes the main steps of our procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In particular, the algorithm employs a procedure RM, an al- gorithm for the full feedback setting as the one described in the previous section, that exposes an interface with the two operation of a traditional regret minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' During each block τ, the full-information subroutine is used to compute a vector xτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, in most stages of the window Iτ, the de- cision at is computed by feeding xτ to the OCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A few stages are chosen uniformly at random to estimate utilities provided by other feasible sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', exploration phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Af- ter the execution of all the stages in the window Iτ, the al- gorithm computes estimated reward functions and uses them to update the full-information regret minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let p : [m] → E be a random permutation of elements in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, for each e ∈ E, by letting j be the index such that p(j) = e in the current block τ, an unbiased estimator ˜fτ(e) of fτ(e) := 1 |Iτ | � t∈Iτ f(1e, wt) can be easily obtained by setting ˜fτ(e) := f(1e, wtj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, it is possible to show that our algorithm provides the following guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a temporal packing feasibility set Fd, and an α-competitive OCRS ˆπ, let Z = T 2/3, and the full feedback subroutine RM be defined as per Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then Algorithm 3 guarantees that α max S∈Id T � t=1 f(1S, wt) − E � T � t=1 f(at, wt) � ≤ ˜O(T 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 6 Conclusion and Future Work In this paper we introduce fully dynamic online selection problems in which selected items affect the combinatorial constraints during their activity times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We presented a gen- eralization of the OCRS approach that provides near opti- mal competitive ratios in the full-information model, and no- α-regret algorithms with polynomial per-iteration running time with both full- and semi-bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Our frame- work opens various future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For example, it would be particularly interesting to understand whether a variation of Algorithms 2 and 3 can be extended to the case in which the adversary changes the constraint family at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Moreover, the study of the bandit-feedback model re- mains open, and no regret bound is known for that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Acknowledgements The authors of Sapienza are supported by the Meta Re- search grant on “Fairness and Mechanism Design”, the ERC Advanced Grant 788893 AMDROMA “Algorithmic and Mechanism Design Research in Online Markets”, the MIUR PRIN project ALGADIMAR “Algorithms, Games, and Dig- ital Markets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' References Abernethy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Hazan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Sanghavi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' and Shakkottai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Blocking Bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Beygelz- imer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=" d'Alch´e-Buc, F." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In Proceedings of the 22nd National Conference on Artificial Intelligence - Volume 1, AAAI’07, 58–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' ISBN 9781577353232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Kesselheim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' and Mehlhorn, K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', 80(2–3): 245–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Kleinberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' and Weinberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Matroid prophet 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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' and Szepesvari, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Tight regret bounds for stochastic combinatorial semi- bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In Artificial Intelligence and Statistics, 535–543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Livanos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A Simple and Tight Greedy OCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' CoRR, abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='13253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' McMahan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' and Blum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Online geometric optimization in the bandit setting against an adaptive adver- sary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In International Conference on Computational Learn- ing Theory, 109–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Orabona, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A modern introduction to online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='13213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A Contention Resolution Schemes and Online Contention Resolution Schemes As explained at length in Section 2, our goal in general is that of finding the independent set of maximum weight for a given feasibility constraint family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' However, doing this directly might be intractable in general and we need to aim for a good approximation of the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In particular, given a non-negative submodular function f : [0, 1]m → R≥0, and a family of packing constraints F, we start from an ex ante feasible solution to the linear program maxx∈PF f(x), which upper bounds the optimal value achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An ex ante feasible solution is simply a distribution over the independent sets of F, given by a vector x in the packing constraint polytope of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' A key observation is that we can interpret the ex ante optimal solution to the above linear program as a vector x∗ of fractional values, which induces distribution over elements such that x∗ e is the marginal probability that element e ∈ E is included in the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, we use this solution to obtain a feasible solution that suitably approximates the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The random set R(x∗) constructed by ex ante selecting each element independently with probability x∗ e can be infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Contention Resolution Schemes (Chekuri, Vondr´ak, and Zenklusen 2011) are procedures that, starting from the random set of sampled elements R(x∗), construct a feasible solution with good approximation guarantees with respect to the optimal solution of the original integer linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Definition 6 (Contention Resolution Schemes (CRSs) (Chekuri, Vondr´ak, and Zenklusen 2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For b, c ∈ [0, 1], a (b, c)- balanced Contention Resolution Scheme (CRS) π for F = (E, I) is a procedure such that, for every ex-ante feasible solution x ∈ bPF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', the down-scaled version of polytope PF), and every subset S ⊆ E, returns a random set π(x, S) ⊆ S satisfying the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Feasibility: π(x, S) ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' c-balancedness: Prπ,R(x) [e ∈ π(x, R(x)) | e ∈ R(x)] ≥ c, ∀e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' When elements arrive in an online fashion, Feldman, Svensson, and Zenklusen (2016) extend CRS to the notion of OCRS, where R(x) is obtained in the same manner, but elements are revealed one by one in adversarial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The procedure has to decide irrevocably whether or not to add the current element to the final solution set, which needs to be feasible and a competitive against the offline optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The idea is that adding a sampled element e ∈ E to the set of already selected elements S ⊆ R(x) maintains feasibility with at least constant probability, regardless of the element and the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This originates Definition 3 and the subsequent discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' B Examples In this section, we provide some clarifying examples for the concepts introduced in Section 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Polytopes Example 4 provides the definition of the constraint polytopes of some standard problems, while Example 5 describes their temporal version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For a set S ⊆ E and x ∈ Rm, we define, with a slight abuse of notation, x(S) := � e∈S xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Example 4 (Standard Polytopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Given a ground set E, Let K = (E, I) be a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, given budget B > 0 and a vector of elements’ sizes c ∈ Rm ≥0, its feasibility polytope is defined as PK = {x ∈ [0, 1]m : ⟨c, x⟩ ≤ B} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let G = (E, I) be a matching constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, its feasibility polytope is defined as PG = {x ∈ [0, 1]m : x(δ(u)) ≤ 1, ∀u ∈ V } , where δ(u) denotes the set of all adjacent edges to u �� V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Note that the ground set in this case is the set of all edges of graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let M = (E, I) be a matroid constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, its feasibility polytope is defined as PM = {x ∈ [0, 1]m : x(S) ≤ rank(S), ∀S ⊆ E} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Here, rank(S) := max {|I| : I ⊆ S, I ∈ I}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=', the cardinality of the maximum independent set contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We can now rewrite the above polytopes under temporal packing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Example 5 (Temporal Polytopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For ground set E, Let K = (E, I) be a knapsack constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, for B > 0 and cost vector c ∈ Rm ≥0, its feasibility polytope is defined as Pd K = {x ∈ [0, 1]m : ⟨c, x⟩ ≤ B, ∀e ∈ E} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let G = (E, I) be a matching constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, its feasibility polytope is defined as Pd G = {x ∈ [0, 1]m : x(δ(u) ∩ Ee) ≤ 1, ∀u ∈ V, ∀e ∈ E} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let M = (E, I) be a matroid constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, its feasibility polytope is defined as Pd M = {x ∈ [0, 1]m : x(S ∩ Ee) ≤ rank(S), ∀S ⊆ E, ∀e ∈ E} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We also note that, for general packing constraints, if de = ∞ for all e ∈ E, then Ee = E, P∞ F = PF, and similarly for the constraint family F∞ = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' From Standard OCRS to Temporal OCRS for Rank-1 Matroids, Matchings, Knapsacks, and General Matroids In this section, we explicitly derive a (1, 1/e)-selectable (randomized) temporal greedy OCRS for the rank-1 matroid feasibility constraint, from a (1, 1/e)-selectable (randomized) greedy OCRS in the standard setting (Livanos 2021), which is also tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let us denote this standard OCRS as πM, where M is a rank-1 matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For the rank-1 matroid feasibility constraint family under temporal constraints, Algorithm 1 produces a (1, 1/e)- selectable (randomized) temporal greedy OCRS ˆπM from πM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Since it is clear from context, we drop the dependence on M and write π, ˆπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We will proceed by comparing side-by-side what happens in π and in ˆπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Let us recall from Examples 4, 5 that the polytopes can respectively be written as PM = {x ∈ [0, 1]m : x(S) ≤ 1, ∀S ⊆ E} , Pd M = {y ∈ [0, 1]m : y(S ∩ Ee) ≤ 1, ∀S ⊆ E, ∀e ∈ E} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The two OCRSs perform the following steps, on the basis of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' On one hand, π defines a subfamily of constraints Fπ,x := {{e} : e ∈ H(x)}, where e ∈ E is included in random subset H(x) ⊆ E with probability 1−e−xe xe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' Then, it selects the first sampled element e ∈ R(x) such that {e} ∈ Fπ,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' On the other hand, πy defines a subfamily of constraints Fd π,y := {{e} : e ∈ H(y)}, where e ∈ E is included in random subset H(y) ⊆ E with probability qe(y) = 1−e−ye ye .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' The feasibility family Fd π,y induces, as per Observation 1, a sequence of feasibility families Fπ,y(e) := {{e} : e ∈ H(y) ∩ Ee}, for each e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' For all e′ ∈ E, the OCRS selects the first sampled element e ∈ R(y) such that {e} ∈ Fπ,y(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In other words, the temporal OCRS selects a sampled element that is active only if no other element in its active elements set has been selected earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' It is clear that both are randomized greedy OCRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' We will now proceed by showing that each element e is selected with probability at least 1/e in both π, ˆπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' In π element e is selected if sampled, and no earlier element has been selected before (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' its singleton set belongs to the subfamily Fπ,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' An element e′ is not selected with probability 1 − xe′ · 1−e−xe′ xe′ = e−xe′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE1T4oBgHgl3EQfVQQy/content/2301.03099v1.pdf'} +page_content=' This means that the probability of e being selected is 1 − e−xe xe � se′