Understanding the Correlation Gap For Matchings

Abstract

Given a set of vertices V with |V| = n, a weight vector w in (R^+ \cup {0})^{\binom{V}{2}}, and a probability vector x In [0, 1]^{\binom{V}{2}} in the matching polytope, we study the quantity (\E_{G}[ \nu_w(G)])/(sum_(u, v) in \binom{V}{2} w_{u, v} x_{u, v}) where G is a random graph where each edge e with weight w_e appears with probability x_e independently, and let \nu_w(G) denotes the weight of the maximum matching of G. This quantity is closely related to correlation gap and contention resolution schemes, which are important tools in the design of approximation  algorithms, algorithmic game theory, and stochastic optimization. 

We provide lower bounds for the above quantity for general and bipartite graphs, and for weighted and unweighted settings. The best known upper bound is 0.54 by Karp and Sipser, and the best lower bound is 0.4. We show that it is at least 0.47 for unweighted bipartite graphs, at least 0.45 for weighted bipartite graphs, and at least 0.43 for weighted general graphs. To achieve our results, we construct local distribution schemes on the dual  which may be of independent interest.