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Approximation Schemes for Minimum 2-Connected Spanning Subgraphs in Weighted Planar Graphs

  • André Berger
  • Artur Czumaj
  • Michelangelo Grigni
  • Hairong Zhao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)

Abstract

We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs, which augments any connected subgraph A of a weighted planar graph G to a (1+ε)-spanner of G with total weight bounded by weight(A)/ε. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have 1 or 2 connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2.

Keywords

Planar Graph Approximation Scheme Jordan Curve Span Subgraph Connectivity Problem 
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 2005

Authors and Affiliations

  • André Berger
    • 1
  • Artur Czumaj
    • 2
  • Michelangelo Grigni
    • 1
  • Hairong Zhao
    • 2
  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.Department of Computer ScienceNew Jersey Institute of TechnologyNewarkUSA

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