Selberg integrals in 1D random Euclidean optimization problems

TL;DR

This paper derives exact average costs for 1D Euclidean optimization problems, like assignment and TSP, using Selberg integrals, providing precise solutions for randomly chosen points.

Contribution

It introduces a novel application of Selberg integrals to compute exact average costs in 1D Euclidean optimization problems.

Findings

Exact average cost for random assignment problem derived.

Exact average cost for bipartite TSP obtained.

Applicable for any number of points and power p.

Abstract

We consider a set of Euclidean optimization problems in one dimension, where the cost function associated to the couple of points $x$ and $y$ is the Euclidean distance between them to an arbitrary power $p\ge1$, and the points are chosen at random with flat measure. We derive the exact average cost for the random assignment problem, for any number of points, by using Selberg's integrals. Some variants of these integrals allows to derive also the exact average cost for the bipartite travelling salesman problem.