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Dynamic Programming for H-minor-free Graphs

  • Juanjo Rué
  • Ignasi Sau
  • Dimitrios M. Thilikos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

We provide a framework for the design and analysis of dynamic programming algorithms for H-minor-free graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2 O(k·logk)·n O(1) steps, with n being the number of vertices of the input graph. Extending the approach developed by the same authors for graphs embedded in surfaces, we introduce a new type of branch decomposition for H-minor-free graphs, called an H-minor-free cut decomposition, and we show that they can be constructed in O h (n 3) steps, where the hidden constant depends exclusively on H. We show that the separators of such decompositions have connected packings whose behavior can be described in terms of a combinatorial object called ℓ-triangulation. Our main result is that when applied on H-minor-free cut decompositions, dynamic programming runs in \(2^{O_h(k)}\cdot n^{O(1)}\) steps. This broadens substantially the class of problems that can be solved deterministically in single-exponential time for H-minor-free graphs.

Keywords

analysis of algorithms parameterized algorithms graphs minors branchwidth dynamic programming non-crossing partitions 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Juanjo Rué
    • 1
  • Ignasi Sau
    • 2
  • Dimitrios M. Thilikos
    • 3
  1. 1.Instituto de Ciencias MatemáticasMadridSpain
  2. 2.AlGCo project-team, CNRSLIRMMMontpellierFrance
  3. 3.Department of MathematicsNational and Kapodistrian University of AthensGreece

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