Moment generating functions in combinatorial optimization: Bipartite matching

TL;DR

This paper studies the distribution of minimum cost bipartite matchings with exponential edge costs, showing it can be efficiently characterized by a rational moment generating function, and explores its zeros for insights into distribution regularity.

Contribution

It introduces an efficient method to compute the distribution of optimal matching costs via the moment generating function and proposes a conjecture about the zeros of this function.

Findings

Distribution represented by rational moment generating function

Zeros of the function relate to distribution regularity

Conjecture on zeros implying Gaussian limit

Abstract

In a random model of minimum cost bipartite matching based on exponentially distributed edge costs, we show that the distribution of the cost of the optimal solution can be computed efficiently. The distribution is represented by its moment generating function, which in this model is always a rational function. The complex zeros of this function are of interest as the lack of zeros near the origin indicates a certain regularity of the distribution. We propose a conjecture according to which these moment generating functions never have complex zeros of smaller modulus than their first pole. For minimum cost perfect matching, also known as the assignment problem, such a zero-free disk would imply a Gaussian scaling limit.