Matroid Secretary for Regular and Decomposable Matroids

In the matroid secretary problem we are given a stream of elements 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 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 matroid classes. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving an O(1)-competitive algorithm for regular matroids (the class of matroids that are representable over every field), and use techniques that are matroid-theoretic rather than graph-theoretic. We use the regular matroid decomposition theorem of Seymour to decompose any regular matroid into matroids which are either graphic, cographic, or isomorphic to R_{10}, and then show how to combine algorithms for these basic classes into an algorithm for regular matroids. 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.Comment: 21 page

The matroid secretary problem for minor-closed classes and random matroids

We prove that for every proper minor-closed class $M$ of matroids representable over a prime field, there exists a constant-competitive matroid secretary algorithm for the matroids in $M$. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle. We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is $(2+o(1))$-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a $(1+o(1))$-competitive algorithm for almost all matroids.Comment: 15 pages, 0 figure

Advances on Matroid Secretary Problems: Free Order Model and Laminar Case

The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constant-factor approximation was found. We address this point by presenting a 9-approximation for the \emph{free order model}, a model suggested shortly after the introduction of the matroid secretary problem, and for which no constant-factor approximation was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed. Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, a constant-factor approximation has been found for this case, using a clever but rather involved method and analysis (Im and Wang, [SODA 2011]) that leads to a 16000/3-approximation. This is arguably the most involved special case of the matroid secretary problem for which a constant-factor approximation is known. We present a considerably simpler and stronger $3\sqrt{3}e\approx 14.12$-approximation, based on reducing the problem to a matroid secretary problem on a partition matroid

The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems

We study the matroid secretary problems with submodular valuation functions. In these problems, the elements arrive in random order. When one element arrives, we have to make an immediate and irrevocable decision on whether to accept it or not. The set of accepted elements must form an {\em independent set} in a predefined matroid. Our objective is to maximize the value of the accepted elements. In this paper, we focus on the case that the valuation function is a non-negative and monotonically non-decreasing submodular function. We introduce a general algorithm for such {\em submodular matroid secretary problems}. In particular, we obtain constant competitive algorithms for the cases of laminar matroids and transversal matroids. Our algorithms can be further applied to any independent set system defined by the intersection of a {\em constant} number of laminar matroids, while still achieving constant competitive ratios. Notice that laminar matroids generalize uniform matroids and partition matroids. On the other hand, when the underlying valuation function is linear, our algorithm achieves a competitive ratio of 9.6 for laminar matroids, which significantly improves the previous result.Comment: preliminary version appeared in STACS 201

The Submodular Secretary Problem Goes Linear

During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. Partially linked to its numerous applications in mechanism design, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular matroid secretary problem (SMSP). So far, O(1)-competitive algorithms have been obtained for SMSP over some basic matroid classes. This created some hope that, analogously to the matroid secretary conjecture, one may even obtain O(1)-competitive algorithms for SMSP over any matroid. However, up to now, most questions related to SMSP remained open, including whether SMSP may be substantially more difficult than MSP; and more generally, to what extend MSP and SMSP are related. Our goal is to address these points by presenting general black-box reductions from SMSP to MSP. In particular, we show that any O(1)-competitive algorithm for MSP, even restricted to a particular matroid class, can be transformed in a black-box way to an O(1)-competitive algorithm for SMSP over the same matroid class. This implies that the matroid secretary conjecture is equivalent to the same conjecture for SMSP. Hence, in this sense SMSP is not harder than MSP. Also, to find O(1)-competitive algorithms for SMSP over a particular matroid class, it suffices to consider MSP over the same matroid class. Using our reductions we obtain many first and improved O(1)-competitive algorithms for SMSP over various matroid classes by leveraging known algorithms for MSP. Moreover, our reductions imply an O(loglog(rank))-competitive algorithm for SMSP, thus, matching the currently best asymptotic algorithm for MSP, and substantially improving on the previously best O(log(rank))-competitive algorithm for SMSP

Simple Random Order Contention Resolution for Graphic Matroids with Almost no Prior Information

Random order online contention resolution schemes (ROCRS) are structured online rounding algorithms with numerous applications and links to other well-known online selection problems, like the matroid secretary conjecture. We are interested in ROCRS subject to a matroid constraint, which is among the most studied constraint families. Previous ROCRS required to know upfront the full fractional point to be rounded as well as the matroid. It is unclear to what extent this is necessary. Fu, Lu, Tang, Turkieltaub, Wu, Wu, and Zhang (SOSA 2022) shed some light on this question by proving that no strong (constant-selectable) online or even offline contention resolution scheme exists if the fractional point is unknown, not even for graphic matroids. In contrast, we show, in a setting with slightly more knowledge and where the fractional point reveals one by one, that there is hope to obtain strong ROCRS by providing a simple constant-selectable ROCRS for graphic matroids that only requires to know the size of the ground set in advance. Moreover, our procedure holds in the more general adversarial order with a sample setting, where, after sampling a random constant fraction of the elements, all remaining (non-sampled) elements may come in adversarial order.Comment: To be published in SOSA2

O(log log Rank) competitive ratio for the Matroid Secretary Problem

In the Matroid Secretary Problem (MSP), the elements of the ground set of a Matroid are revealed on-line one by one, each together with its value. An algorithm for the MSP is called Matroid-Unknown if, at every stage of its execution, it only knows (i) the elements that have been revealed so far and their values and (ii) an oracle for testing whether or not a subset the elements that have been revealed so far forms an independent set. An algorithm is called Known-Cardinality if it knows (i), (ii) and also knows from the start the cardinality n of the ground set of the Matroid. We present here a Known-Cardinality algorithm with a competitive-ratio of order log log the rank of the Matroid. The prior known results for a OC algorithm are a competitive-ratio of log the rank of the Matroid, by Babaioff et al. (2007), and a competitive-ratio of square root of log the rank of the Matroid, by Chakraborty and Lachish (2012)