PTAS for Steiner Tree on Map Graphs
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 nodeweighted Steiner tree problem, for which only a 2.4approximation 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 nodeweighted Steiner tree to finding a spanner, i.e., a constantfactor approximation containing a nearly optimum solution. Finally, we pinpoint places where known techniques for constructing such spanner fail on node weighted instances and further progress requires new ideas.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.00717
 Bibcode:
 2019arXiv191200717B
 Keywords:

 Computer Science  Data Structures and Algorithms