Towards the distribution of the smallest matching in the Random Assignment Problem

TL;DR

This paper explores the distribution of the minimum matching cost in the random assignment problem, proposing conjectures that, if proven, would describe its asymptotic distribution and confirming results for a special case.

Contribution

It introduces conjectures on the distributional properties of matchings and establishes the limiting distribution for a specific case of the problem.

Findings

Conjectures on the distributional properties of matchings.

Proven limiting distribution for a special case.

Predicted asymptotic normality of the centered minimum cost.

Abstract

We consider the problem of minimizing cost among one-to-one assignments of $n$ jobs onto $n$ machines. The random assignment problem refers to the case when the cost associated with performing jobs on machines are random variables. Aldous established the expected value of the smallest cost, $A_n$, in the limiting $n$ regime. However the distribution of the minimum cost has not been established yet. In this paper we conjecture some distributional properties of matchings in matrices. If this conjecture is proved, this will establish that $\sqrt{n}(A_n - E(A_n)) \overset{w}{\Rightarrow} N(0,2)$. We also establish the limiting distribution for a special case of the Random Assignment Problem.