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Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Branch Decompositions

  • Frederic Dorn
  • Eelko Penninkx
  • Hans L. Bodlaender
  • Fedor V. Fomin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)

Abstract

Divide-and-conquer strategy based on variations of the Lipton-Tarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with new techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. Compared to divide-and-conquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an  \(O(2^{6.903\sqrt{n}}n^{3/2}+n^{3})\) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time \(O(2^{10.8224\sqrt{n}}n^{3/2}+n^{3})\). Our approach can be used to design parameterized algorithms as well. For example we introduce the first \(2^{O\sqrt{k}}k^{O(1)}.n^{O(1)}\) time algorithm for parameterized Planar k –cycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length ≥ k in time \(O(2^{13.6\sqrt{k}}\sqrt{k}n+n^{3})\).

Keywords

Planar Graph Travel Salesman Problem Exact Algorithm Hamiltonian Cycle Vertex Cover 
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

  • Frederic Dorn
    • 1
  • Eelko Penninkx
    • 2
  • Hans L. Bodlaender
    • 2
  • Fedor V. Fomin
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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