Calculus of Variations and Geometric Measure Theory

D. Trevisan - M. Goldman

Optimal transport methods for combinatorial optimization over two random point sets

created by goldman on 30 Sep 2022


Submitted Paper

Inserted: 30 sep 2022
Last Updated: 30 sep 2022

Year: 2022


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.