Rapidly Mixing Markov Chain Monte Carlo Technique for Matching Problems with Global Utility Function

TL;DR

This paper introduces a Markov Chain Monte Carlo method for solving a complex bipartite matching problem with a utility function, proving rapid mixing and polynomial convergence despite NP-hardness.

Contribution

It presents a randomized MCMC algorithm for the problem and proves the chain's rapid mixing property with a conductance bound, a novel theoretical result.

Findings

NP-hardness of the matching problem established

Constructed Markov chain is reversible and positive recurrent

Proved polynomial mixing time via conductance bounds

Abstract

This paper deals with a complete bipartite matching problem with the objective of finding an optimal matching that maximizes a certain generic predefined utility function on the set of all matchings. After proving the NP-hardness of the problem using reduction from the 3-SAT problem, we propose a randomized algorithm based on Markov Chain Monte Carlo (MCMC) technique for solving this. We sample from Gibb's distribution and construct a reversible positive recurrent discrete time Markov chain (DTMC) that has the steady state distribution same as the Gibb's distribution. In one of our key contributions, we show that the constructed chain is `rapid mixing', i.e., the convergence time to reach within a specified distance to the desired distribution is polynomial in the problem size. The rapid mixing property is established by obtaining a lower bound on the conductance of the DTMC graph and this result is of independent interest.