Calculus of Variations and Geometric Measure Theory

M. Goldman - D. Trevisan

Optimal transport methods for combinatorial optimization over two random point sets

created by goldman on 30 Sep 2022
modified by trevisan on 05 Jul 2024


Accepted Paper

Inserted: 30 sep 2022
Last Updated: 5 jul 2024

Journal: PTRF
Year: 2022

ArXiv: 2209.14615 PDF


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.