Theory of Computing Systems

, Volume 47, Issue 3, pp 637–654 | Cite as

Trimmed Moebius Inversion and Graphs of Bounded Degree

  • Andreas Björklund
  • Thore Husfeldt
  • Petteri Kaski
  • Mikko Koivisto
Article

Abstract

We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an n-element universe U and a family ℱ of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of U with k sets from ℱ in time within a polynomial factor (in n) of the number of supersets of the members of ℱ.

Relying on an projection theorem of Chung et al. (J. Comb. Theory Ser. A 43:23–37, 1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs with maximum degree Δ. In particular, we show how to compute the domatic number in time within a polynomial factor of (2Δ+1−2)n/(Δ+1) and the chromatic number in time within a polynomial factor of (2Δ+1−Δ−1)n/(Δ+1). For any constant Δ, these bounds are O((2−ε)n) for ε>0 independent of the number of vertices n.

Keywords

Graph algorithms Inclusion-exclusion Chromatic number Domatic number 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Andreas Björklund
    • 1
  • Thore Husfeldt
    • 1
    • 2
  • Petteri Kaski
    • 3
  • Mikko Koivisto
    • 3
  1. 1.Department of Computer ScienceLund UniversityLundSweden
  2. 2.IT University of CopenhagenKøbenhavn SDenmark
  3. 3.Department of Computer ScienceHelsinki Institute for Information Technology HIIT, University of HelsinkiHelsinkiFinland

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