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.
Progress has been made on the matroid secretary conjecture for variants of which modify the assumptions on the order in which elements arrive and the way weights are assigned to elements. One simpler variant of is obtained by assuming . Here, an adversary can only choose @math not necessarily distinct weights, and the weights are assigned to the elements uniformly at random. In this model, a @math -competitive algorithm can be obtained for any matroid @cite_19 . Additionally, a @math -competitive procedure can still be obtained in the random weight assignment model even if elements are assumed to arrive in an adversarial order @cite_30 @cite_19 . Hence, this variant is, in a sense, the opposite of the classical , where weights are adversarial and the arrival order is random. Furthermore, a @math -competitive algorithm can be obtained in the so-called . Here, the weight assignment is adversarial; however, the algorithm can choose the order in which elements arrive @cite_9 @cite_11 . Among the above-discussed variants, this is the only variant with adversarial weight assignments for which an @math -competitive algorithm is known. For more information on recent advances on and its variants we refer to the survey @cite_20 .
@cite_30 @cite_9 @cite_19 @cite_20 @cite_11
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The classical secretary problem studies the problem of hiring the best secretary from among the secretaries who arrive in random order by making immediate and irrevocable decisions. After the interesting connection to online mechanism design was found [19, 20], the random order input assumption has been studied for a variety of problems. [4] formalized a general version of the secretary problem, namely the matroid secretary problem. In the problem, a secretary corresponds to an element in the universe U. The goal is to select the maximum weight independent set. They conjectured that the matroid secretary problem, for any matroid, allows a constant competitive algorithm. The conjecture remains open. Some constant approximation algorithms are currently known for some special cases of matroids. Another interesting type of secretary problem was studied where elements have non-uniform sizes, as is the case in the knapsack secretary problem [3, 6]. In this paper, we consider two interesting secretary problems. One is when the matroid is a laminar matroid, which generalizes uniform partition truncated partition matroids. For the laminar matroid secretary problem, using a novel replacement rule which we call "kick next," we give the first constant-competitive algorithm. The other is the interval scheduling secretary problem, which generalizes the knapsack secretary problem. In this problem, each job Ji arrives with interval Ii, processing time pi and weight wi. If Ji is accepted, then it must be scheduled during Ii, not necessarily continuously. The goal is to accept the jobs of the maximum total weight which are schedulable. We give a simple O(log D)-competitive algorithm and a nearly matching lower bound on the competitive ratio of any randomized algorithm, where D is the maximum interval length of any job. 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 , 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 @math -approximation, based on reducing the problem to a matroid secretary problem on a partition matroid. The Matroid Secretary Problem, introduced by (2007), is a generalization of the Classical Secretary Problem. In this problem, elements from a matroid are presented to an on-line algorithm in a random order. Each element has a weight associated with it, which is revealed to the algorithm along with the element. After each element is revealed the algorithm must make an irrevocable decision on whether or not to select it. The goal is to pick an independent set with the sum of the weights of the selected elements as large as possible. gave an algorithm for the Matroid Secretary Problem with a competitive ratio of O(logρ), where ρ is the rank of the matroid. It has been conjectured that a constant competitive-ratio is achievable for this problem. In this paper we give an algorithm that has a competitive-ratio of O(√logρ). We study a generalization of the classical secretary problem which we call the "matroid secretary problem". In this problem, the elements of a matroid are presented to an online algorithm in random order. When an element arrives, the algorithm observes its value and must make an irrevocable decision regarding 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. This paper presents 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.