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Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

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Abstract

We present a general 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 and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  \(O(2^{6.903\sqrt{n}})\) time algorithm solving weighted Hamiltonian Cycle on an n-vertex planar graph. Similar technique solves Planar Graph Travelling Salesman Problem with n cities in time \(O(2^{9.8594\sqrt{n}})\) . Our approach can be used to design parameterized algorithms as well. For example, we give an algorithm that for a given k decides if a planar graph on n vertices has a cycle of length at least k in time \(O(2^{13.6\sqrt{k}}n+n^{3})\) .

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, 5020, Bergen, Norway

    Frederic Dorn & Fedor V. Fomin

  2. Department of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB, Utrecht, The Netherlands

    Eelko Penninkx & Hans L. Bodlaender

Authors
  1. Frederic Dorn
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  2. Eelko Penninkx
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  3. Hans L. Bodlaender
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  4. Fedor V. Fomin
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Corresponding author

Correspondence to Frederic Dorn.

Additional information

This work is supported by the Norwegian Research Council and partially by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation).

A preliminary version of this paper appeared at ALGO-ESA’05 [15].

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Dorn, F., Penninkx, E., Bodlaender, H.L. et al. Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions. Algorithmica 58, 790–810 (2010). https://doi.org/10.1007/s00453-009-9296-1

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  • DOI: https://doi.org/10.1007/s00453-009-9296-1

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