Primal-Dual Approximation Algorithms for Node-Weighted Network Design in Planar Graphs

We present primal-dual algorithms which give a 2.4 approximation for a class of node-weighted network design problems in planar graphs, introduced by Demaine, Hajiaghayi and Klein (ICALP’09). This class includes Node-Weighted Steiner Forest problem studied recently by Moldenhauer (ICALP’11) and other nodeweighted problems in planar graphs that can be expressed using (0, 1)-proper functions introduced by Goemans and Williamson. We show that these problems can be equivalently formulated as feedback vertex set problems and analyze approximation factors guaranteed by different violation oracles within primal-dual framework developed for such problems by Goemans and Williamson. For edge-weighted versions of these problems, as well as versions with uniform node weights, there has been significant progress in obtaining PTASs recently. There were proposed general techniques, such as bidimensionality of Demaine and Hajiaghayi, algorithmic theory of vertex separators of Feige, Hajiaghayi and Lee and contraction decomposition of Demaine, Hajiaghayi and Kawarabayashi. For Edge-Weighted Steiner Forest PTAS was obtained by Bateni, Hajiaghayi and Marx and by Eisenstat, Klein and Mathieu. In contrast, for more general node-weighted versions constant factor approximations via primal-dual algorithms remain the state of the art, while no APX-hardness is known.