Minor-free graphs have light spanners

TL;DR

This paper proves that all $H$-minor-free graphs possess light $(1+ ext{epsilon})$-spanners with bounds depending on the minor's size, resolving longstanding open problems and improving TSP approximation schemes.

Contribution

It establishes the existence of light spanners in $H$-minor-free graphs with bounds depending on the minor's size, resolving open problems and conjectures.

Findings

Existence of light $(1+ ext{epsilon})$-spanners in $H$-minor-free graphs.

Bound on lightness depending on $rac{ ext{size of } H}{ ext{epsilon}^3} imes ext{log}$ factor.

Implication for efficient PTAS for TSP in $H$-minor-free graphs.

Abstract

We show that every $H$-minor-free graph has a light $(1+\epsilon)$-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung. Our lightness bound is \[O\left(\frac{\sigma_H}{\epsilon^3}\log \frac{1}{\epsilon}\right)\] where $\sigma_H = |V(H)|\sqrt{\log |V(H)|}$ is the sparsity coefficient of $H$-minor-free graphs. That is, it has a practical dependency on the size of the minor $H$. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in $H$-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is $2^{O_H\left(\frac{1}{\epsilon^4}\log \frac{1}{\epsilon}\right)}n^{O(1)}$ where $O_H$ ignores dependencies on the size of $H$. Our techniques significantly deviate from existing lines of research on spanners for $H$-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs.