Understanding the Correlation Gap for Matchings

TL;DR

This paper investigates the correlation gap for matchings, providing improved lower bounds for various graph classes, which enhances understanding of approximation limits in stochastic matching problems.

Contribution

The paper establishes new lower bounds for the correlation gap in general, bipartite, weighted, and unweighted graphs, improving upon previous bounds and introducing novel local distribution schemes.

Findings

Lower bound of 0.47 for unweighted bipartite graphs

Lower bound of 0.45 for weighted bipartite graphs

Lower bound of 0.43 for weighted general graphs

Abstract

Given a set of vertices $V$ with $|V| = n$, a weight vector $w \in (\mathbb{R}^+ \cup \{ 0 \})^{\binom{V}{2}}$, and a probability vector $x \in [0, 1]^{\binom{V}{2}}$ in the matching polytope, we study the quantity $\frac{E_{G}[ \nu_w(G)]}{\sum_{(u, v) \in \binom{V}{2}} w_{u, v} x_{u, v}}$ 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. he 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 lea st $0.43$ for weighted general graphs. To achieve our results, we construct local distribution schemes on the dual which may be of independent interest.