The Stochastic Matching Problem with (Very) Few Queries

Abstract

Motivated by an application in kidney exchange, we study the following stochastic matching problem: we are given a graph G(V, E) (not necessarily bipartite), where each edge in E is realized with some constant probability p > 0 and the goal is to find a maximum matching in the realized graph. An algorithm in this setting is allowed to make queries to edges in E in order to determine whether or not they are realized. We design an adaptive algorithm for this problem that, for any graph G, computes a (1 − ε)-approximate maximum matching in the realized graph Gp with high probability, while making O log (1/εp) εp queries per vertex, where the edges to query are chosen adaptively in O log (1/εp) εp rounds. We further present a non-adaptive algorithm that makes O log (1/εp) εp queries per vertex and computes a (1 2 − ε)-approximate maximum matching in Gp with high probability. Both our adaptive and non-adaptive algorithms achieve the same approximation factor as the previous best algorithms of Blum et al. (EC 2015) for this problem, while requiring exponentially smaller number of per-vertex queries (and rounds of adaptive queries for the adaptive algorithm). Our results settle an open problem raised by Blum et al. by achieving only a polynomial dependency on both ε and p. Moreover, the approximation guarantee of our algorithms is instance-wise as opposed to only being competitive in expectation as is the case for Blum et al.. This is of particular relevance to the key application of stochastic matching in kidney exchange. We obtain our results via two main techniques, namely matching-covers and vertex sparsification that may be of independent interest.

Stochastic Packing Integer Programs with Few Queries

“…Our stochastic packing integer programming problem generalizes the stochastic (unweighted) matching problem [3,4,7] and the stochastic (unweighted) k-hypergraph matching problem [7], which have recently been studied in EC (Economics and Computation) community. These problems are motivated to find an optimal strategy for kidney exchange [16,33].…”

“…For the stochastic unweighted matching problem, Blum et al [7] proposed adaptive and non-adaptive algorithms that achieve approximation ratios of (1 − ) and of ( [3] proposed adaptive and non-adaptive algorithms that respectively achieve the same approximation ratios with high probability, by conducting O(log(1/ p)/ p) queries per vertex. Their technique is based on the Tutte-Berge formula and vertex sparsification.…”

“…Relationship to the analysis of Assadi et al The adaptive and non-adaptive algorithms of Assadi et al [3] for the unweighted non-bipartite matching problem are within our framework (Algorithms 1 and 2, respectively), and their analysis utilizes the TutteBerge formula. They showed that the required number of iterations is O(log(n/ µ)/ p), and it is reduced to O (poly(p, 1/ )) by using the vertex sparsification lemma.…”

“…The second part is a simple extension of Assadi et al [3]. Since they only analyzed this case, ν = ω(1) was required.…”

“…In particular, our and Assadi et al [3]'s procedures are similar to the one in [11], which is aimed to for reducing space complexity of packing problems in streaming setting, but these conduct different analysis and provide different guarantees.…”