The Number of Optimal Matchings for Euclidean Assignment on the Line

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 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 S_N.