O(log log Rank) Competitive Ratio for the Matroid Secretary Problem

Lachish, Oded

doi:10.1109/focs.2014.42

Cited by 71 publications

(72 citation statements)

References 18 publications

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“…Chakraborty and Lachish [CL12] gave an improved O( √ log ρ)-competitive algorithm, which was further improved to O(log log ρ) by Lachish [Lac14] and by Feldman, Svensson, and Zenklusen [FSZ14]. Some other classes of matroids have been shown to admit O(1)-competitive algorithms (e.g.…”

Section: Introductionmentioning

confidence: 98%

Matroid Secretary for Regular and Decomposable Matroids

Dinitz¹,

Kortsarz²

2013

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

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In the matroid secretary problem we are given a stream of elements in random order and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get rid of it, and if we choose not to accept it then we cannot later add it. Babaioff, Immorlica, and Kleinberg [SODA 2007] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)-competitive algorithm. However, most matroids that are known to admit an O(1)-competitive algorithm can be easily represented using graphs (e.g. graphic, cographic, and transversal matroids). In particular, there is very little known about F -representable matroids (the class of matroids that can be represented as elements of a vector space over a field F ), which are one of the foundational types of matroids. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving O(1)-competitive algorithms for regular matroids (the class of matroids that are representable over any field), and use techniques that are fundamentally matroid-theoretic rather than graphtheoretic.Our main technique is to leverage the seminal regular matroid decomposition theorem of Seymour, which gives a method for decomposing any regular matroid into matroids which are either graphic, cographic, or isomorphic to a simple 10-element matroid. We show how to combine in a black-box manner any algorithms for these basic classes into an algorithm for a given regular matroid, i.e. how to respect the decomposition. In fact, this allows us to generalize beyond regular matroids to any class of matroids that admits such a decomposition into classes for which we already have good algorithms. In particular, we give an O(1)-competitive algorithm for the class of max-flow min-cut matroids, which Seymour showed can be decomposed into regular matroids and copies of the Fano matroid.

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Section: Introductionmentioning

confidence: 98%

Matroid Secretary for Regular and Decomposable Matroids

Dinitz¹,

Kortsarz²

2013

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

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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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“…This line of research has received significant attention recently, in part due to applications to auction theory and mechanism design. The most famous variant is the matroid secretary problem, where the chosen subset forms an independent set in a matroid [5,24,13], but recent work considered more general combinatorial constraints as well [28]. Secretary settings with non-uniform random arrival orders have been investigated in [21].…”

Section: Related Workmentioning

confidence: 99%

Stable Secretaries

Babichenko

Emek

Feldman

et al. 2017

Proceedings of the 2017 ACM Conference on Economics and Computation

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We define and study a new variant of the secretary problem. Whereas in the classic setting multiple secretaries compete for a single position, we study the case where the secretaries arrive one at a time and are assigned, in an on-line fashion, to one of multiple positions. Secretaries are ranked according to talent, as in the original formulation, and in addition positions are ranked according to attractiveness. To evaluate an online matching mechanism, we use the notion of blocking pairs from stable matching theory: our goal is to maximize the number of positions (or secretaries) that do not take part in a blocking pair. This is compared with a stable matching in which no blocking pair exists. We consider the case where secretaries arrive randomly, as well as that of an adversarial arrival order, and provide corresponding upper and lower bounds.

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O(log log Rank) Competitive Ratio for the Matroid Secretary Problem

Cited by 71 publications

References 18 publications

Matroid Secretary for Regular and Decomposable Matroids

Matroid Secretary for Regular and Decomposable Matroids

Prophet Secretary for Combinatorial Auctions and Matroids

Stable Secretaries

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