Primal-Dual Approximation Algorithms for Node-Weighted Steiner Forest on Planar Graphs

  • Carsten Moldenhauer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


Node-Weighted Steiner Forest is the following problem: Given an undirected graph, a set of pairs of terminal vertices, a weight function on the vertices, find a minimum weight set of vertices that includes and connects each pair of terminals. We consider the restriction to planar graphs where the problem remains NP-complete. Demaine et al. [DHK09] showed that the generic primal-dual algorithm of Goemans and Williamson [GW97] is a 6-approximation on planar graphs. We present (1) a different analysis to prove an approximation factor of 3, (2) show that this bound is tight for the generic algorithm, and (3) show how the algorithm can be improved to yield a 9/4-approximation algorithm.

We give a simple proof for the first result using contraction techniques and present an example for the lower bound. Then, we establish a connection to the feedback problems studied by Goemans and Williamson [GW98]. We show how our constructions can be combined with their proof techniques yielding the third result and an alternative, more involved, way of deriving the first result. The third result induces an upper bound on the integrality gap of 9/4. Our analysis implies that improving this bound for Node-Weighted Steiner Forest via the primal-dual algorithm is essentially as difficult as improving the integrality gap for the feedback problems in [GW98].


Planar Graph STEINER Tree Steiner Tree Problem Terminal Vertex Black Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Carsten Moldenhauer
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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