A PTAS for subset TSP in minor-free graphs

TL;DR

This paper presents the first polynomial-time approximation scheme (PTAS) for the subset TSP problem in minor-free graphs, advancing the understanding of efficient routing in complex graph families.

Contribution

It introduces a novel PTAS for subset TSP in minor-free graphs using nearly light subset spanners based on sparse spanner oracles, resolving a long-standing open problem.

Findings

First PTAS for subset TSP in minor-free graphs.

Constructed nearly light subset spanners using sparse spanner oracles.

Improved lightness bounds for spanners in doubling and constant correlation dimension metrics.

Abstract

We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in $H$-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in minor-closed families initiated by Grigni, Koutsoupias and Papadimitriou in FOCS'95. The main technical ingredient in our PTAS is a construction of a nearly light subset $(1+\epsilon)$-spanner for any given edge-weighted $H$-minor-free graph. This construction is based on a necessary and sufficient condition given by \emph{sparse spanner oracles}: light subset spanners exist if and only if sparse spanner oracles exist. This relationship allows us to obtain two new results: _ An $(1+\epsilon)$-spanner with lightness $O(\epsilon^{-d+2})$ for any doubling metric of constant dimension $d$. This improves the earlier lightness bound $\epsilon^{-O(d)}$ obtained by Borradaile, Le and Wulff-Nilsen. _ An $(1+\epsilon)$-spanner with sublinear lightness for any metric of constant correlation dimension. Previously, no spanner with non-trivial lightness was known.