Dynamic Programming for Graphs on Surfaces

  • Juanjo Rué
  • Ignasi Sau
  • Dimitrios M. Thilikos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

Abstract

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in \(2^{\mathcal{O}(k\cdot \log k)}\cdot n\) steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called surface cut decomposition, capturing how partial solutions can be arranged on a surface. Then we use singularity analysis over expressions obtained by the symbolic method to prove that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in \(2^{\mathcal{O}(k)}\cdot n\) steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve all previous results in this direction.

Keywords

analysis of algorithms parameterized algorithms analytic combinatorics graphs on surfaces branchwidth dynamic programming polyhedral embeddings symbolic method non-crossing partitions 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Juanjo Rué
    • 1
  • Ignasi Sau
    • 2
  • Dimitrios M. Thilikos
    • 3
  1. 1.Laboratorie d’InformatiqueÉcole PolytechniquePalaiseau-CedexFrance
  2. 2.Department of Computer ScienceTechnionHaifaIsrael
  3. 3.Dept. of MathematicsNational and Kapodistrian University of AthensGreece

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