Light spanners for bounded treewidth graphs imply light spanners for $H$-minor-free graphs

TL;DR

This paper explores the relationship between light spanners in bounded treewidth graphs and H-minor-free graphs, aiming to support a conjecture that could improve approximation algorithms for the traveling salesperson problem.

Contribution

It establishes that light greedy spanners in bounded pathwidth graphs imply the conjecture for H-minor-free graphs, advancing understanding of graph spanners and approximation schemes.

Findings

Greedy spanner of bounded pathwidth graphs is light.

If bounded treewidth graphs have light greedy spanners, then H-minor-free graphs do too.

Discussion on extending results to bounded treewidth graphs.

Abstract

Grigni and Hung~\cite{GH12} conjectured that H-minor-free graphs have $(1+\epsilon)$-spanners that are light, that is, of weight $g(|H|,\epsilon)$ times the weight of the minimum spanning tree for some function $g$. This conjecture implies the {\em efficient} polynomial-time approximation scheme (PTAS) of the traveling salesperson problem in $H$-minor free graphs; that is, a PTAS whose running time is of the form $2^{f(\epsilon)}n^{O(1)}$ for some function $f$. The state of the art PTAS for TSP in H-minor-free-graphs has running time $n^{1/\epsilon^c}$. We take a further step toward proving this conjecture by showing that if the bounded treewidth graphs have light greedy spanners, then the conjecture is true. We also prove that the greedy spanner of a bounded pathwidth graph is light and discuss the possibility of extending our proof to bounded treewidth graphs.