Towards an Optimal Contention Resolution Scheme for Matchings
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In this paper, we study contention resolution schemes for matchings. Given a fractional matching x and a random set R(x) where each edge e appears independently with probability x_e, we want to select a matching M ⊆ R(x) such that [e ∈ M | e ∈ R(x)] ≥ c, for c as large as possible. We call such a selection method a c-balanced contention resolution scheme. Our main results are (i) an asymptotically (in the limit as x_∞ goes to 0) optimal ≃ 0.544-balanced contention resolution scheme for general matchings, and (ii) a 0.509-balanced contention resolution scheme for bipartite matchings. To the best of our knowledge, this result establishes for the first time, in any natural relaxation of a combinatorial optimization problem, a separation between (i) offline and random order online contention resolution schemes, and (ii) monotone and non-monotone contention resolution schemes. We also present an application of our scheme to a combinatorial allocation problem, and discuss some open questions related to van der Waerden's conjecture for the permanent of doubly stochastic matrices.
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