Trimmed Moebius Inversion and Graphs of Bounded Degree

Authors:
Andreas Björklund;Thore Husfeldt;Petteri Kaski;Mikko Koivisto
Affiliations:
Lund University, Department of Computer Science, P.O. Box 118, 22100, Lund, Sweden;Lund University, Department of Computer Science, P.O. Box 118, 22100, Lund, Sweden and IT University of Copenhagen, Rued Langgaards Vej 7, 2300, København S, Denmark;Helsinki Institute for Information Technology HIIT, University of Helsinki, Department of Computer Science, P.O. Box 68, 00014, Helsinki, Finland;Helsinki Institute for Information Technology HIIT, University of Helsinki, Department of Computer Science, P.O. Box 68, 00014, Helsinki, Finland
Venue:
Theory of Computing Systems - Special Title: Symposium on Theoretical Aspects of Computer Science; Guest Editors: Susanne Albers, Pascal Weil
Year:
2010

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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 ε0 independent of the number of vertices n.