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 Soto11 ). This is under the assumption that the matroid is known in advance. If the matroid is unknown in advance, we present an @math -approximation, and prove that a better than @math approximation is impossible. This resolves an open question posed by BIK07 . As a natural special case, we also consider the classical secretary problem where the number of candidates @math is unknown in advance. If @math is chosen by an adversary from @math , we provide a nearly tight answer, by providing an algorithm that chooses the best candidate with probability at least @math and prove that a probability better than @math cannot be achieved (where @math is the @math -th harmonic number).
The primary variant of matroid secretary problem (RO-AA-MN model) was introduced in @cite_6 . In the following, let @math denote the total number of elements and @math the rank of the matroid. An @math -approximation for the RO-AA-MN model was given in @cite_6 . It was also conjectured that a constant-factor approximation should exist for this problem and this question is still open. Very recently, Chakraborty and Lachish @cite_13 improved @cite_6 by giving an @math -approximation algorithm. Constant-factor approximations were given in @cite_6 for some special cases such as partition matroids and graphic matroids with a given explicit representation. Further, constant-factor approximations were given for transversal matroids @cite_21 @cite_22 and laminar matroids @cite_18 . However, even for graphic matroids in the RO-AA-MK model when the graphic matroid is given by an oracle, no constant factor is known.
@cite_18 @cite_22 @cite_21 @cite_6 @cite_13
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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 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. 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. 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).