  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.
