Prize-collecting Steiner problems on planar graphs

  • Authors:

  • M. Bateni;C. Chekuri;A. Ene;M. T. Hajiaghayi;N. Korula;D. Marx

  • Affiliations:

  • Princeton University, Princeton, NJ;University of Illinois, Urbana, IL;University of Illinois, Urbana, IL;University of Maryland;Google Research, New York, NY;Humboldt-Universität zu Berlin, Germany

  • Venue:
  • Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
  • Year:
  • 2011

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Abstract

In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF), and more generally Submodular Prize-Collecting Steiner Forest (SPCSF), on planar graphs (and also on bounded-genus graphs) to the corresponding problem on graphs of bounded treewidth. More precisely, for each of the mentioned problems, an α-approximation algorithm for the problem on graphs of bounded treewidth implies an (α + ∈)-approximation algorithm for the problem on planar graphs (and also bounded-genus graphs), for any constant ∈ 0. PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth and hence we obtain a PTAS for these problems on planar graphs and bounded-genus graphs. In contrast, we show that PCSF is APX-hard to approximate on series-parallel graphs, which are planar graphs of treewidth at most 2. Apart from ruling out a PTAS for PCSF on planar graphs and bounded treewidth graphs, this result is also interesting since it gives the first provable hardness separation between the approximability of a problem and its prize-collecting version. We also show that PCSF is APX-hard on Euclidean instances.