  We obtain polynomial-time approximation-preserving reductions (up to a factor
of 1 + \epsilon) from the prize-collecting Steiner tree and prize-collecting
Steiner forest problems in planar graphs to the corresponding problems in
graphs of bounded treewidth. We also give an exact algorithm for the
prize-collecting Steiner tree problem that runs in polynomial time for graphs
of bounded treewidth. This, combined with our reductions, yields a PTAS for the
prize-collecting Steiner tree problem in planar graphs and generalizes the PTAS
of Borradaile, Klein and Mathieu for the Steiner tree problem in planar graphs.
Our results build upon the ideas of Borradaile, Klein and Mathieu and the work
of Bateni, Hajiaghayi and Marx on a PTAS for the Steiner forest problem in
planar graphs. Our main technical result is on the properties of primal-dual
algorithms for Steiner tree and forest problems in general graphs when they are
run with scaled up penalties.
