Prize-Collecting Steiner Tree and Forest in Planar Graphs

We obtain polynomial-time approximation-preserving reductions (up to a factor of 1+ε) from the prizecollecting 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 [11] for the Steiner tree problem in planar graphs. Our results build upon the ideas in [11] and the work of Bateni, Hajiaghayi and Marx [8] 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.