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.
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This research was supported in part by the Academy of Finland, Grants 117499 (P.K.) and 109101 (M.K.), and by the Swedish Research Council, project “Exact Algorithms” (A.B. and T.H.).
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Björklund, A., Husfeldt, T., Kaski, P. et al. Trimmed Moebius Inversion and Graphs of Bounded Degree. Theory Comput Syst 47, 637–654 (2010). https://doi.org/10.1007/s00224-009-9185-7
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DOI: https://doi.org/10.1007/s00224-009-9185-7