Minor-free graphs have light spanners

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 \[O\left(\frac{σ_H}{ε^3}\log \frac{1}ε\right)\] 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.

22 pages, 4 figures. Accepted to FOCS 2017