### When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings [chapter]

Nikhil Bansal, Anupam Gupta, Jian Li, Julián Mestre, Viswanath Nagarajan, Atri Rudra
2010 Lecture Notes in Computer Science
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
more » ... ximize 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  . 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. : 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 different rounding procedures to attain the bounds claimed above. • The first algorithm ( §2.1) is very simple: it considers edges in a uniformly random order and probes each edge e with probability proportional to y e ; the analysis uses Markov's inequality and a Chernofftype bound (Lemma 2). • The second algorithm ( §2.2) is more nuanced and achieves a better bound: we use dependent rounding  on the y-values to obtain a setÊ of edges to be probed, and then probe edges ofÊ in a uniformly random order. Though the first algorithm has a weaker approximation ratio, we still present it since it is useful in the online stochastic matching problem (Section 3). The second rounding algorithm has an additional advantage: The probing strategy returned by the algorithm can be made matching-probing  . In this alternative (more restrictive) probing model we are given an additional parameter k and edges need to be probed in k rounds, each round being a matching. It is clear that this matching-probing model is more restrictive than the usual edge-probing model (with timeouts min{t i , k}) where one edge is probed at a time. Our algorithm obtains a matching-probing strategy that is only a small constant factor worse than the optimal edge-probing strategy; hence, we also obtain the same constant approximation guarantee for weighted stochastic matching in the matching-probing model. It is worth noting that previously only a logarithmic approximation in the unweighted case was known  . Theorem 2 There is a 4-approximation algorithm for the weighted stochastic matching problem in the matching-probing model. For bipartite graphs, there is a 3-approximation algorithm. Notice that for general graphs our algorithm matches the performance of the greedy algorithm shown by Chen et al.  for the unweighted case. Interestingly, even though their individual analyses show that they are 4-approximations, they can be combined to obtain better approximations. Theorem 3 There is a 3.46-approximation algorithm for the unweighted stochastic matching problem in general graphs. Apart from solving these open problems and yielding improved approximations, our LP-based analysis turns out to be applicable in a wider context.