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

Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the writeup on Fixed-Threshold algorithm

Improved algorithms and analysis for the laminar matroid secretary problem

In a matroid secretary problem, one is presented with a sequence of objects of various weights in a random order, and must choose irrevocably to accept or reject each item. There is a further constraint that the set of items selected must form an independent set of an associated matroid. Constant-competitive algorithms (algorithms whose expected solution weight is within a constant factor of the optimal) are known for many types of matroid secretary problems. We examine the laminar matroid and show an algorithm achieving provably 0.053 competitive ratio

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

Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints

We study various generalizations of the secretary problem with submodular objective functions. Generally, a set of requests is revealed step-by-step to an algorithm in random order. For each request, one option has to be selected so as to maximize a monotone submodular function while ensuring feasibility. For our results, we assume that we are given an offline algorithm computing an alpha-approximation for the respective problem. This way, we separate computational limitations from the ones due to the online nature. When only focusing on the online aspect, we can assume alpha = 1. In the submodular secretary problem, feasibility constraints are cardinality constraints, or equivalently, sets are feasible if and only if they are independent sets of a k-uniform matroid. That is, out of a randomly ordered stream of entities, one has to select a subset of size k. For this problem, we present a 0.31alpha-competitive algorithm for all k, which asymptotically reaches competitive ratio alpha/e for large k. In submodular secretary matching, one side of a bipartite graph is revealed online. Upon arrival, each node has to be matched permanently to an offline node or discarded irrevocably. We give a 0.207alpha-competitive algorithm. This also covers the problem, in which sets of entities are feasible if and only if they are independent with respect to a transversal matroid. In both cases, we improve over previously best known competitive ratios, using a generalization of the algorithm for the classic secretary problem. Furthermore, we give an O(alpha d^(-2/(B-1)))-competitive algorithm for submodular function maximization subject to linear packing constraints. Here, d is the column sparsity, that is the maximal number of none-zero entries in a column of the constraint matrix, and B is the minimal capacity of the constraints. Notably, this bound is independent of the total number of constraints. We improve the algorithm to be O(alpha d^(-1/(B-1)))-competitive if both d and B are known to the algorithm beforehand

Stochastic Combinatorial Optimization via Poisson Approximation

We study several stochastic combinatorial problems, including the expected utility maximization problem, the stochastic knapsack problem and the stochastic bin packing problem. A common technical challenge in these problems is to optimize some function of the sum of a set of random variables. The difficulty is mainly due to the fact that the probability distribution of the sum is the convolution of a set of distributions, which is not an easy objective function to work with. To tackle this difficulty, we introduce the Poisson approximation technique. The technique is based on the Poisson approximation theorem discovered by Le Cam, which enables us to approximate the distribution of the sum of a set of random variables using a compound Poisson distribution. We first study the expected utility maximization problem introduced recently [Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we obtain an additive PTAS if there is a multidimensional PTAS for the multi-objective version of the problem, strictly generalizing the previous result. For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and Tardos, STOC97]), we show there is a polynomial time algorithm which uses at most the optimal number of bins, if we relax the size of each bin and the overflow probability by eps. For stochastic knapsack, we show a 1+eps-approximation using eps extra capacity, even when the size and reward of each item may be correlated and cancelations of items are allowed. This generalizes the previous work [Balghat, Goel and Khanna, SODA11] for the case without correlation and cancelation. Our algorithm is also simpler. We also present a factor 2+eps approximation algorithm for stochastic knapsack with cancelations. the current known approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of the 45th ACM Symposium on the Theory of Computing (STOC13