Chapter

Algorithms - ESA 2009

Volume 5757 of the series Lecture Notes in Computer Science pp 554-565

Inclusion/Exclusion Meets Measure and Conquer

Exact Algorithms for Counting Dominating Sets
  • Johan M. M. van RooijAffiliated withDepartment of Information and Computing Sciences, Utrecht University
  • , Jesper NederlofAffiliated withDepartment of Informatics, University of Bergen
  • , Thomas C. van DijkAffiliated withDepartment of Information and Computing Sciences, Utrecht University

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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.