The number of optimal matchings for Euclidean Assignment on the line

TL;DR

This paper analyzes the number and distribution of optimal matchings in the 1D Euclidean Assignment Problem with linear cost, providing formulas and asymptotic behavior for the entropy of the solution space.

Contribution

It characterizes all optimal matchings, derives a formula for their count, and studies the asymptotic distribution of the entropy in the random ensemble.

Findings

Number of optimal matchings grows as exp(S_N) with S_N ~ (1/2) N log N + N s.

Distribution p(s) of the entropy's random component is independent of N.

Provides asymptotic expansions for moments of p(s) using Brownian process and singularity analysis.

Abstract

We consider the Random Euclidean Assignment Problem in dimension $d=1$, with linear cost function. In this version of the problem, in general, there is a large degeneracy of the ground state, i.e. there are many different optimal matchings (say, $\sim \exp(S_N)$ at size $N$). We characterize all possible optimal matchings of a given instance of the problem, and we give a simple product formula for their number. Then, we study the probability distribution of $S_N$ (the zero-temperature entropy of the model), in the uniform random ensemble. We find that, for large $N$, $S_N \sim \frac{1}{2} N \log N + N s + \mathcal{O}\left( \log N \right)$, where $s$ is a random variable whose distribution $p(s)$ does not depend on $N$. We give expressions for the asymptotics of the moments of $p(s)$, both from a formulation as a Brownian process, and via singularity analysis of the generating functions associated to $S_N$. The latter approach provides a combinatorial framework that allows to compute an asymptotic expansion to arbitrary order in $1/N$ for the mean and the variance of