Concentration for random Euclidean combinatorial optimization

TL;DR

This paper establishes concentration bounds for random Euclidean combinatorial optimization problems, such as bipartite matching and TSP, in higher dimensions, using a novel geometric and probabilistic approach.

Contribution

It introduces a new method combining Poincaré inequalities with geometric bounds to prove concentration for $p$-costs in Euclidean optimization problems.

Findings

Concentration at the natural energy scale $n^{1-p/d}$ for $1 extless p extless d^2/2$.

Method successfully applies to bipartite matching and TSP in dimensions $d extgreater 2$.

Proposes a conjectural transfer principle to extend results to all $p extgreater 1$.

Abstract

We prove concentration bounds for random Euclidean combinatorial optimization problems with $p$--costs. For bipartite matching and for the (mono- and bi-partite) traveling salesperson problem in dimension $d\ge 3$, we obtain concentration at the natural energy scale $n^{1-p/d}$ for $1\le p<d^2/2$. Our method combines a Poincar\'e inequality with a robust geometric mechanism providing uniform bounds on the edges of optimizers. We also formulate a conjectural $p\!\to\!q$ transfer principle for the $p$--optimal matching which, if true, would extend the concentration range to all $p\ge 1$.