Stochastic Packing Integer Programs with Few Queries

Maehara, Takanori; Yamaguchi, Yutaro

doi:10.1137/1.9781611975031.21

Cited by 17 publications

(21 citation statements)

References 23 publications

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“…Interestingly, as pointed out in Reference [32], the proposed strategy in Reference [32] coincides with our approach when applied to the stochastic matching problem. However, while the results in Reference [32] span a wider set of problems, the bounds achieved in Reference [32] for the stochastic matching problem are weaker than ours by a factor of Θ(log n), i.e., Reference [32] achieves algorithms with O (…”

Section: Subsequent Worksupporting

confidence: 68%

“…Our techniques in this article were also extended to achieve efficient adaptive and non-adaptive algorithms for the more general problem of stochastic integer programs [32]. Interestingly, as pointed out in Reference [32], the proposed strategy in Reference [32] coincides with our approach when applied to the stochastic matching problem.…”

Section: Subsequent Workmentioning

confidence: 57%

“…Subsequent to the conference version of this article [7], the stochastic matching problem has been studied further in References [8,12,32]. In a follow-up work [8], we were able to break the 1/2 approximation barrier for the non-adaptive case in previous work [7,13] and achieve a non-adaptive algorithm with approximation ratio strictly larger than 1/2.…”

Section: Subsequent Workmentioning

confidence: 97%

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The Stochastic Matching Problem with (Very) Few Queries

AssadiSepehr

KhannaSanjeev

Liyang

2019

ACM Trans. Econ. Comput.

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Motivated by an application in kidney exchange, we study the following stochastic matching problem: We are given a graph G (V , E) (not necessarily bipartite), where each edge in E is realized with some constant probability p > 0, and the goal is to find a maximum matching in the realized graph. An algorithm in this setting is allowed to make queries to edges in E to determine whether or not they are realized. We design an adaptive algorithm for this problem that, for any graph G, computes a (1 − ε)-approximate maximum matching in the realized graph G p with high probability, while making O (log (1/εp) εp) queries per vertex, where the edges to query are chosen adaptively in O (log (1/εp) εp) rounds. We further present a nonadaptive algorithm that makes O (log (1/εp) εp) queries per vertex and computes a (1 2 − ε)-approximate maximum matching in G p with high probability. Both our adaptive and non-adaptive algorithms achieve the same approximation factor as the previous best algorithms of Blum et al. (EC 2015) for this problem, while requiring an exponentially smaller number of per-vertex queries (and rounds of adaptive queries for the adaptive algorithm). Our results settle an open problem raised by Blum et al. by achieving only a polynomial dependency on both ε and p. Moreover, the approximation guarantee of our algorithms is instance-wise as opposed to only being competitive in expectation as is the case for Blum et al. This is of particular relevance to the key application of stochastic matching in kidney exchange. We obtain our results via two main techniques-namely, matching-covers and vertex sparsification-that may be of independent interest.

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Section: Subsequent Worksupporting

confidence: 68%

Section: Subsequent Workmentioning

confidence: 57%

Section: Subsequent Workmentioning

confidence: 97%

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The Stochastic Matching Problem with (Very) Few Queries

AssadiSepehr

KhannaSanjeev

Liyang

2019

ACM Trans. Econ. Comput.

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“…The stochastic matching problem has been intensively studied during the past decade due to its diverse applications from kidney exchange to labor markets and online dating (we overview these applications in Section 1.1). Directly related to the setting that we consider are the papers by Blum [YM18], and Behnezhad and Reyhani [BR18]. Table 1 gives a brief survey of known results due to these papers as well as a comparison to our results.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, if ϕ(N ) ≥ opt/2, the fractional matching obtained by Lemma 4.7 which also satisfies blossom inequalities, implies that an integral matching of size at least (1 − 10 )opt/2 must exist in the realization. Note that Corollary 4.8 already improves the number of per-vertex queries of known results for weighted graphs due to [BR18,YM18]. Our goal, however, is to provide a much better guarantee on the approximation factor.…”

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confidence: 96%

Stochastic Matching with Few Queries: New Algorithms and Tools

Behnezhad¹,

Farhadi²,

Hajiaghayi³

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the following stochastic matching problem on both weighted and unweighted graphs: A graph G = (V, E) along with a parameter p ∈ (0, 1) is given in the input. Each edge of G is realized independently with probability p. The goal is to select a degree bounded (dependent only on p) subgraph H of G such that the expected size/weight of maximum realized matching of H is close to that of G.This model of stochastic matching has attracted significant attention over the recent years due to its various applications in kidney exchange, online labor markets, and other matching markets.The most fundamental open question is the best approximation factor achievable for such algorithms that, in the literature, are referred to as non-adaptive algorithms. Prior work has identified breaking (near) half-approximation as a barrier for both weighted and unweighted graphs. Our main results are as follows:• We analyze a simple and clean algorithm and show that for unweighted graphs, it finds an (almost) 4 √ 2 − 5 (≈ 0.6568) approximation by querying O( log(1/p) p ) edges per vertex. This improves over the state-of-the-art 0.5001 approximation of Assadi et al. [EC'17].• We show that the same algorithm achieves a 0.501 approximation for weighted graphs by querying O( log(1/p) p ) edges per vertex. This is the first algorithm to break 0.5 approximation barrier for weighted graphs. It also improves the per-vertex queries of the state-ofthe-art by Yamaguchi and Maehara [SODA'18] and Behnezhad and Reyhani [EC'18].Prior results were all interestingly based on similar algorithms and differed only in the analysis. Our algorithms are fundamentally different, yet very simple and natural. For the analysis, we introduce a number of procedures that construct heavy fractional matchings. We consider the new algorithms and our analytical tools to be the main contributions of this paper. * Portion of the work was completed while some of the authors were at

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