A Simple <i>O</i>(log log(rank))-Competitive Algorithm for the Matroid Secretary Problem

Feldman, Moran; Svensson, Ola; Zenklusen, Rico

doi:10.1287/moor.2017.0876

Cited by 21 publications

(12 citation statements)

References 11 publications

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“…Meanwhile, the connection between secretary problems and online auctions is first explored in Hajiaghayi et al [25]. Its generalization to matroids is considered in [5,18,38] and to matchings in [22,24,27,31,35,40].…”

Section: Related Workmentioning

confidence: 99%

Prophet Secretary for Combinatorial Auctions and Matroids

Ehsani¹,

Hajiaghayi²,

Keßelheim³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

108

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The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). and Feldman et al. [17] show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/2-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/2-approximation and obtain (1 − 1/e)-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan [45] and Esfandiari et al. [15] who worked in the special cases where we can fully control the arrival order or when there is only a single item.Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

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Section: Related Workmentioning

confidence: 99%

Prophet Secretary for Combinatorial Auctions and Matroids

Ehsani¹,

Hajiaghayi²,

Keßelheim³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

108

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show abstract

“…For general matroids, there has been sequence of improving competitive ratio with the state of the art being O(log log r) [43,27]. Obtaining a constant competitive ratio for general matroids remains a central open problem, but constant bounds are known for many special cases (see survey by Dinitz [19] and references therein).…”

Section: Further Related Workmentioning

confidence: 99%

Combinatorial Prophet Inequalities

Rubinstein¹,

Singla²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downwardclosed feasibility constraint, we give an O(log n log 2 r)-competitive algorithm (where r is the cardinality of the largest feasible subset).Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log 2 r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an Ω( √ n) lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model.En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for nonmonotone submodular functions.

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“…However, there is no constant ratio for the general submodular matroid secretary problem till now. Feldman and Zenklusen [FZ15] reduced the problem to the matroid secretary problem with linear objective functions, which implies an O(log log(rank))-competitive algorithm for the submodular matroid secretary problem, matching the current best result for the matroid secretary problem [Lac14,FSZ15].…”

Section: K-uniformmentioning

confidence: 99%

Online Submodular Maximization with Free Disposal: Randomization Beats ¼ for Partition Matroids

Chan¹,

Huang²,

Jiang³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access.For k-uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches . We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased.Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1 α∞ ≈ 0.3178.Finally, our techniques can be extended to a more general problem that generalizes both the online submodular maximization problem and the online bipartite matching problem with free disposal. Using the techniques developed in this paper, we give constantcompetitive algorithms for the submodular online bipartite matching problem.

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A Simple O(log log(rank))-Competitive Algorithm for the Matroid Secretary Problem

Cited by 21 publications

References 11 publications

Prophet Secretary for Combinatorial Auctions and Matroids

Prophet Secretary for Combinatorial Auctions and Matroids

Combinatorial Prophet Inequalities

Online Submodular Maximization with Free Disposal: Randomization Beats ¼ for Partition Matroids

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