Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

TL;DR

This paper develops polynomial-time approximation schemes for key subset-connectivity problems in bounded-genus graphs, extending previous planar graph results to a broader class of graphs with similar efficiency.

Contribution

It introduces the first PTASes for Steiner tree, survivable-network design, and subset TSP in bounded-genus graphs, generalizing planar graph techniques.

Findings

PTAS algorithms run in O(n log n) time.

Extends planar graph PTAS frameworks to bounded-genus graphs.

Applicable to both orientable and non-orientable surfaces.

Abstract

We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and non-orientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu from planar graphs to bounded-genus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to bounded-genus graphs.