The stochastic matching problem deals with finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic k-set packing, where the problem is to find a maximum packing of sets, each of which exists with some probability. In this paper, we provide edge and set query algorithms for these two problems, respectively, that provably achieve some fraction of the omniscient optimal solution. Our main theoretical result for the stochastic matching (i.e., 2-set packing) problem is the design of an adaptive algorithm that queries only a constant number of edges per vertex and achieves a (1 − ) fraction of the omniscient optimal solution, for an arbitrarily small > 0. Moreover, this adaptive algorithm performs the queries in only a constant number of rounds. We complement this result with a non-adaptive (i.e., one round of queries) algorithm that achieves a (0.5 − ) fraction of the omniscient optimum. We also extend both our results to stochastic k-set packing by designing an adaptive algorithm that achieves a ( − for the deterministic k-set packing problem [Fürer and Yu 2013] . We empirically explore the application of (adaptations of) these algorithms to the kidney exchange problem, where patients with end-stage renal failure swap willing but incompatible donors. We show on both generated data and on real data from the first 169 match runs of the UNOS nationwide kidney exchange that even a very small number of non-adaptive edge queries per vertex results in large gains in expected successful matches.
More specifically, Blum et al. REF give an adaptive algorithm that achieves a (1 − ϵ)-approximation by querying log(2/ϵ ) p 2/ϵ pervertex edges, in the same number of rounds.
Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries
EC '15
Computer Science