Separating subadditive Euclidean functionals

TL;DR

This paper proves that the constants associated with the asymptotic lengths of various Euclidean functionals like TSP, MST, and matching are strictly separated, providing insights into their relative sizes and computational complexity.

Contribution

It establishes strict inequalities among the asymptotic constants for Euclidean TSP, MST, matching, and 2-factor, and separates the TSP from its LP relaxation asymptotically.

Findings

Proves $eta_{MST}^d<eta_{TSP}^d$ for all $d",

Shows $2eta_{MM}^d<eta_{TSP}^d$ for all $d",

Separates TSP from its LP relaxation asymptotically.

Abstract

If we are given $n$ random points in the hypercube $[0,1]^d$, then the minimum length of a Traveling Salesperson Tour through the points, the minimum length of a spanning tree, and the minimum length of a matching, etc., are known to be asymptotically $\beta n^{\frac{d-1}{d}}$ a.s., where $\beta$ is an absolute constant in each case. We prove separation results for these constants. In particular, concerning the constants $\beta_{\mathrm{TSP}}^d$, $\beta_{\mathrm{MST}}^d$, $\beta_{\mathrm{MM}}^d$, and $\beta_{\mathrm{TF}}^d$ from the asymptotic formulas for the minimum length TSP, spanning tree, matching, and 2-factor, respectively, we prove that $\beta_{\mathrm{MST}}^d<\beta_{\mathrm{TSP}}^d$, $2\beta_{\mathrm{MM}}^d<\beta_{\mathrm{TSP}}^d$, and $\beta_{\mathrm{TF}}^d<\beta_{\mathrm{TSP}}^d$ for all $d\geq 2$. We also asymptotically separate the TSP from its linear programming relaxation in this setting. Our results have some computational relevance, showing that a certain natural class of simple algorithms cannot solve the random Euclidean TSP efficiently.