Separating subadditive Euclidean functionals

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.