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Advances on Matroid Secretary Problems: Free Order Model and Laminar Case
, 2012
"... The most wellknown conjecture in the context of matroid secretary problems claims the existence of a constantfactor 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 secreta ..."
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The most wellknown conjecture in the context of matroid secretary problems claims the existence of a constantfactor 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 [19, 17]. However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constantfactor approximation was found. We address this point by presenting a 9approximation for the free order model, a model suggested shortly after the introduction of the matroid secretary problem, and for which no constantfactor 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 constantfactor approximation has been found for this case, using a clever but rather involved method and analysis [12] that leads to a 16000/3approximation. This is arguably the most involved special case of the matroid secretary problem for which a constantfactor approximation is known. We present a considerably simpler and stronger 3 √ 3e ≈ 14.12approximation, based on reducing the problem to a matroid secretary problem on a partition matroid.
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 ..."
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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, cographic, and transversal matroids). In particular, there is very little known about Frepresentable 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
Secretary Problems with Convex Costs
"... We consider online resource allocation problems where given a set of requests our goal is to select a subset that maximizes a value minus cost type of objective function. Requests are presented online in random order, and each request possesses an adversarial value and an adversarial size. The onlin ..."
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We consider online resource allocation problems where given a set of requests our goal is to select a subset that maximizes a value minus cost type of objective function. Requests are presented online in random order, and each request possesses an adversarial value and an adversarial size. The online algorithm must make an irrevocable accept/reject decision as soon as it sees each request. The “profit ” of a set of accepted requests is its total value minus a convex cost function of its total size. This problem generalizes the socalled knapsack secretary problem and is closely related to the submodular secretary problem. Unlike previous work on secretary problems, one of the main challenges we face is that the objective function can be positive or negative and we must guard against accepting requests that look good early on but cause the solution to have an arbitrarily large cost as more requests are accepted. We study this problem under various feasibility constraints and present online algorithms with competitive ratios only a constant factor worse than those known in the absence of costs for the same feasibility constraints. We also consider a multidimensional version of the problem that generalizes multidimensional knapsack within a secretary framework. In the absence of
The simulated greedy algorithm for several submodular matroid secretary problems
 STACS
, 2013
"... 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 ..."
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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 nonnegative and monotonically nondecreasing 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.
Interviewing secretaries in parallel
 In Proceedings of the 13th ACM Conference on Electronic Commerce
, 2012
"... Motivated by the parallel nature of online internet helpdesks and human inspections, we introduce the study of interviewing secretaries in parallel, extending upon the study of the classical secretary problem. In our setting secretaries arrive into multiple queues, and are interviewed in parallel, ..."
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Motivated by the parallel nature of online internet helpdesks and human inspections, we introduce the study of interviewing secretaries in parallel, extending upon the study of the classical secretary problem. In our setting secretaries arrive into multiple queues, and are interviewed in parallel, with the aim of recruiting several secretaries in a timely manner. We consider a variety of new problems that fit this setting, and provide both upper and lower bounds on the efficiency of the corresponding interviewing policies, contrasting them with the classical single queue setting.
Secretary Problems with NonUniform Arrival Order
"... Abstract For a number of problems in the theory of online algorithms, it is known that the assumption that elements arrive in uniformly random order enables the design of algorithms with much better performance guarantees than under worstcase assumptions. The quientessential example of this phenom ..."
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Abstract For a number of problems in the theory of online algorithms, it is known that the assumption that elements arrive in uniformly random order enables the design of algorithms with much better performance guarantees than under worstcase assumptions. The quientessential example of this phenomenon is the secretary problem, in which an algorithm attempts to stop a sequence at the moment it observes the maximum value in the sequence. As is well known, if the sequence is presented in uniformly random order there is an algorithm that succeeds with probability 1/e, whereas no nontrivial performance guarantee is possible if the elements arrive in worstcase order. In many of the applications of online algorithms, it is reasonable to assume there is some randomness in the input sequence, but unreasonable to assume that the arrival ordering is uniformly random. This work initiates an investigation into relaxations of the randomordering hypothesis in online algorithms, by focusing on the secretary problem and asking what performance guarantees one can prove under relaxed assumptions. Toward this end, we present two sets of properties of distributions over permutations as sufficient conditions, called the (p, q, δ)blockindependence property and (k, δ)uniforminducedordering property. We show these two are asymptotically equivalent by borrowing some techniques from the celebrated approximation theory. Moreover, we show they both imply the existence of secretary algorithms with constant probability of correct selection, approaching the optimal constant 1/e as the related parameters of the property tend towards their extreme values. Both of these properties are significantly weaker than the usual assumption of uniform randomness; we substantiate this by providing several constructions of distributions that satisfy (p, q, δ)blockindependence. As one application of our investigation, we prove that Θ(log log n) is the minimum entropy of any permutation distribution that permits constant probability of correct selection in the secretary problem with n elements. While our blockindependence condition is sufficient for constant probability of correct selection, it is not necessary; however, we present complexitytheoretic evidence that no simple necessary and sufficient criterion exists. Finally, we explore the extent to which the performance guarantees of other algorithms are preserved when one relaxes the uniform random ordering assumption to (p, q, δ)blockindependence, obtaining a positive result for Kleinberg's multiplechoice secretary algorithm and a negative result for the weighted bipartite matching algorithm of Korula
Deterministic Protocol for Shared Qqueue Jchoice Kbest Secretary Problem
, 2014
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Maximization of NonMonotone Submodular Functions
, 2014
"... A litany of questions from a wide variety of scientific disciplines can be cast as nonmonotone submodular maximization problems. Since this class of problems includes maxcut, it is NPhard. Thus, generalpurpose algorithms for the class tend to be approximation algorithms. For unconstrained proble ..."
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A litany of questions from a wide variety of scientific disciplines can be cast as nonmonotone submodular maximization problems. Since this class of problems includes maxcut, it is NPhard. Thus, generalpurpose algorithms for the class tend to be approximation algorithms. For unconstrained problem instances, one recent innovation in this vein includes an algorithm of Buchbinder et al. (2012) that guarantees a 1/2approximation to the maximum. Building on this, for problems subject to cardinality constraints, Buchbinder et al. (2014) offer guarantees in the range [0.356, 1/2 + o(1)]. Earlier work has the best approximation factors for more complex constraints and settings. For constraints that can be characterized as a solvable polytope, Chekuri et al. (2011) provide guarantees. For the online secretary setting, Gupta et al. (2010) provide guarantees. In sum, the current body of work on nonmonotone submodular maximization lays