PTAS for Steiner Tree on Map Graphs

TL;DR

This paper presents a Polynomial-Time Approximation Scheme (PTAS) for the Steiner tree problem on map graphs, a class that extends planar graphs, by leveraging contraction decomposition and spanner techniques.

Contribution

It introduces a PTAS for Steiner tree on map graphs and advances understanding of spanner construction challenges in node-weighted planar instances.

Findings

Established a PTAS for Steiner tree on map graphs.

Identified limitations of existing spanner techniques for node-weighted instances.

Connected the problem to planar node-weighted Steiner tree approximations.

Abstract

We study the Steiner tree problem on map graphs, which substantially generalize planar graphs as they allow arbitrarily large cliques. We obtain a PTAS for Steiner tree on map graphs, which builds on the result for planar edge weighted instances of Borradaile et al. The Steiner tree problem on map graphs can be casted as a special case of the planar node-weighted Steiner tree problem, for which only a 2.4-approximation is known. We prove and use a contraction decomposition theorem for planar node weighted instances. This readily reduces the problem of finding a PTAS for planar node-weighted Steiner tree to finding a spanner, i.e., a constant-factor approximation containing a nearly optimum solution. Finally, we pin-point places where known techniques for constructing such spanner fail on node weighted instances and further progress requires new ideas.