Optimal transport methods for combinatorial optimization over two random point sets

TL;DR

This paper studies the asymptotic behavior of the minimum cost for a broad class of combinatorial optimization problems on random bipartite geometric graphs in Euclidean space, extending previous results to a larger parameter range.

Contribution

It establishes almost sure convergence of the normalized minimum cost for these problems in higher dimensions and for larger p, using subadditivity and optimal transport techniques.

Findings

Proves convergence of the scaled minimum cost for p<d in high dimensions.

Extends previous results limited to p<d/2.

Introduces new bounds for Euclidean bipartite matching via optimal transport.

Abstract

We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in $\mathbb{R}^d$ where the edge cost between two points is given by a $p$-th power of their Euclidean distance. This includes e.g.\ the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as $n$ grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and $d \ge 3$, $1\le p<d$. Previous results were limited to the range $p<d/2$. Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.