A PTAS for subset TSP in minor-free graphs

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 sparse spanner oracles : light subset spanners exist if and only if sparse spanner oracles exist. This relationship allows us to obtain two new results: center dot 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 [14]. center dot An (1 + epsilon)-spanner with sublinear lightness for any metric of constant correlation dimension. Previously, no spanner with non-trivial lightness was known.