Matroid Secretary Problem Research Articles

Secretary and online matching problems with machine learned advice

The classic analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. In contrast, machine learning approaches shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take these predictions into account. In particular, we study the following online selection problems: (i) the classic secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classic online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases.

A Framework for the Secretary Problem on the Intersection of Matroids

The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives.

Strong Algorithms for the Ordinal Matroid Secretary Problem

In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: see text] is probability-competitive if every element from the optimum appears with probability [Formula: see text] in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of [Formula: see text] for laminar matroids. Additionally, we modify Kleinberg’s [Formula: see text] utility-competitive algorithm for uniform matroids of rank [Formula: see text] in order to obtain a [Formula: see text] probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.

The Matroid Secretary Problem for Minor-Closed Classes and Random Matroids

We prove that for every proper minor-closed class $\mathcal{M}$ of $\mathbb{F}_p$-representable matroids, there exists an $O(1)$-competitive algorithm for the matroid secretary problem on $\mathcal...

Matroid Secretary Problems

We define a generalization of the classical secretary problem called the matroid secretary problem . In this problem, the elements of a matroid are presented to an online algorithm in uniformly random order. When an element arrives, the algorithm observes its value and must make an irrevocable decision whether or not to accept it. The accepted elements must form an independent set, and the objective is to maximize the combined value of these elements. We present an O (log k )-competitive algorithm for general matroids (where k is the rank of the matroid), and constant-competitive algorithms for several special cases including graphic matroids, truncated partition matroids, and bounded degree transversal matroids. We leave as an open question the existence of constant-competitive algorithms for general matroids. Our results have applications in welfare-maximizing online mechanism design for domains in which the sets of simultaneously satisfiable agents form a matroid.

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

Only recently, progress has been made in obtaining o(log(rank))-competitive algorithms for the matroid secretary problem. More precisely, Chakraborty and Lachish (2012) presented a O((log(rank))1/2)-competitive procedure, and Lachish (2014) later presented a O(log log(rank))-competitive algorithm. Both these algorithms and their analyses are very involved, which is also reflected in the extremely high constants in their competitive ratios. Using different tools, we present a considerably simpler O(log log(rank))-competitive algorithm for the matroid secretary problem. Our algorithm can be interpreted as a distribution over a simple type of matroid secretary algorithms that are easy to analyze. Because of the simplicity of our procedure, we are also able to vastly improve on the hidden constant in the competitive ratio.

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. The interest in MSP is twofold: on the one hand, there are many interesting applications of MSP, and on the other hand, there is strong hope that MSP admits $O(1)$-competitive algorithms, which is the claim of the well-known matroid secretary conjecture. Partially linked to its numerous applications in online auctions, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular MSP (SMSP). The fact that submodularity captures the property of diminishing returns, a very natural property for valuation functions, is a key reason for the interest in 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 questi...

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 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 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 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.

Matroid Secretary for Regular and Decomposable Matroids

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 [Proceedings of SODA'07] 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 graph-theoretic. 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 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.

On Variants of the Matroid Secretary Problem

We present a number of positive and negative results for variants of the matroid secretary problem. Most notably, we design a constant-factor competitive algorithm for the “random assignment” model where the weights are assigned randomly to the elements of a matroid, and then the elements arrive on-line in an adversarial order (extending a result of Soto, SODA 2011, pp. 1275–1284, 2011). This is under the assumption that the matroid is known in advance. If the matroid is unknown in advance, we present an O(logrlogn)-approximation, and prove that a better than O(logn/loglogn) approximation is impossible. This resolves an open question posed by Babaioff et al. (SODA 2007, pp. 434–443, 2007).As a natural special case, we also consider the classical secretary problem where the number of candidates n is unknown in advance. If n is chosen by an adversary from {1,…,N}, we provide a nearly tight answer, by providing an algorithm that chooses the best candidate with probability at least 1/(H N−1+1) and prove that a probability better than 1/H N cannot be achieved (where H N is the N-th harmonic number).

Recent advances on the matroid secretary problem

The matroid secretary problem was introduced by Babaioff, Immorlica, and Kleinberg in SODA 2007 as an online problem that was both mathematically interesting and had applications to online auctions. In this column I will introduce and motivate this problem, and give a survey of some of the exciting work that has been done on it over the past 6 years. While we have a much better understanding of matroid secretary now than we did in 2007, the main conjecture is still open: does there exist an O(1)-competitive algorithm.

Matroid Secretary Problem in the Random-Assignment Model

The matroid secretary problem admits several variants according to the order in which the matroid's elements are presented and how the elements are assigned weights. As the main result of this article, we devise the first constant competitive algorithm for the model in which both the order and the weight assignment are selected uniformly at random, achieving a competitive ratio of approximately $5.7187$. This result is based on the nontrivial fact that every matroid can be approximately decomposed into uniformly dense minors. Based on a preliminary version of this work, Oveis Gharan and Vondrák [Proceedings of the 19th Annual European Symposium on Algorithms, ESA, 2011, pp. 335--346] devised a $40e/(e-1)$-competitive algorithm for the stronger random-assignment adversarial-order model. In this article we present an alternative algorithm achieving a competitive ratio of $16e/(e-1)$. As additional results, we obtain new algorithms for the standard model of the matroid secretary problem: the adversarial-assignment random-order model. We present an $O(\log r)$-competitive algorithm for general matroids which, unlike previous ones, uses only comparisons among seen elements. We also present constant competitive algorithms for various matroid classes, such as column-sparse representable matroids and low-density matroids. The latter class includes, as a special case, the duals of graphic matroids.