Inclusion/Exclusion Meets Measure and Conquer

Exact Algorithms for Counting Dominating Sets
  • Johan M. M. van Rooij
  • Jesper Nederlof
  • Thomas C. van Dijk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)

Abstract

In this paper, two central techniques from the field of exponential time algorithms are combined for the first time: inclusion/exclusion and branching with measure and conquer analysis.

In this way, we have obtained an algorithm that, for each κ, counts the number of dominating sets of size κ in \(\mathcal{O}(1.5048^n)\) time. This algorithm improves the previously fastest algorithm that counts the number of minimum dominating sets. The algorithm is even slightly faster than the previous fastest algorithm for minimum dominating set, thus improving this result while computing much more information.

When restricted to c-dense graphs, circle graphs, 4-chordal graphs or weakly chordal graphs, our combination of branching with inclusion/exclusion leads to significantly faster counting and decision algorithms than the previously fastest algorithms for dominating set.

All results can be extended to counting (minimum) weight dominating sets when the size of the set of possible weight sums is polynomially bounded.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Johan M. M. van Rooij
    • 1
  • Jesper Nederlof
    • 2
  • Thomas C. van Dijk
    • 1
  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Department of InformaticsUniversity of BergenBergenNorway

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