When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

Abstract

Consider a random graph model where each possible edge e is present independently with some probability p e . Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most t i times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are the following:• We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09]. • Combining our LP-rounding algorithm with the natural greedy algorithm, we give an improved 3.46 approximation for unweighted stochastic matching on general graphs. • We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.1 of unsuccessful dates a person might be willing to participate in, "timeouts" on vertices are also provided. More precisely, valid policies are allowed, for each vertex i, to only probe at most t i edges incident to i. Similar considerations arise in kidney exchanges, details of which appear in [7]. Chen et al. asked the question: how can we devise probing policies to maximize the expected cardinality (or weight) of the matching? They showed that the greedy algorithm that probes edges in decreasing order of p ij (as long as their endpoints had not timed out) was a 4-approximation to the cardinality version of the stochastic matching problem. This greedy algorithm (and other simple greedy schemes) can be seen to be arbitrarily bad in the presence of weights, and they left open the question of obtaining good algorithms to maximize the expected weight of the matching produced. In addition to being a natural generalization, weights can be used as a proxy for revenue generated in matchmaking services. (The unweighted case can be thought of as maximizing the social welfare.) In this paper, we resolve the main open question from Chen et al. [7]:Theorem 1 There is a 4-approximation algorithm for the weighted stochastic matching problem. For bipartite graphs, there is a 3-approximation algorithm.Our main idea is to use the knowledge of edge probabilities to solve a linear program where each edge e has a variable 0 ≤ y e ≤ 1 corresponding to the probability that a strategy probes e (over all possible realizations of the graph). This is similar to the approach for stochastic packing problems considered by Dean et al. [9,8]. We then give two d...

An Improved Approximation for k -Median and Positive Correlation in Budgeted Optimization

“…Starting with the key deterministic "pipage rounding" algorithm of Ageev & Sviridenko [1], dependent-rounding schemes have been interpreted probabilistically, and have found several applications and generalizations in combinatorial optimization (see, e.g., [4,6,9,15,17,28] for a small sample). These naturally induce certain types of negative correlation [28] and sometimes even-more powerful negative-association properties [12,22], which are useful in proving Chernoff-like concentration bounds on various monotone functions of such random variables.…”

Online Matching with Stochastic Rewards

“…Another related line of work is that of Chen et al [7] and Bansal et al [3]. They consider an offline matching problem on general random graphs, with query budgets.…”

“…We observe that the offline version of the ONLINE STOCHASTIC MATCHING problem is indeed a special case of this problem. The only online problem considered in this line of work is by Bansal et al [3], who study a hybrid of online stochastic arrivals (which is weaker than the classical online model) and stochastic rewards (similar to our problem), but with multiple trials, and achieve a competitive ratio of of about 0.13.…”

When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

“…An extended abstract of this paper appeared in the Proceedings of the 18th Annual European Symposium on Algorithms [3]. The bounds presented here in §2 are slightly better than those claimed in the extended abstract.…”