Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Björklund, Andreas; Husfeldt, Thore; Kaski, Petteri; Koivisto, Mikko http://www.dagstuhl.de/lipics License
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Trimmed Moebius Inversion and Graphs of Bounded Degree

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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 $scr F$ of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of $U$ with $k$ sets from $scr F$ in time within a polynomial factor (in $n$) of the number of supersets of the members of $scr F$. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs of maximum degree $Delta$. In particular, we show how to compute the Domatic Number in time within a polynomial factor of $(2^{Delta+1-2)^{n/(Delta+1)$ and the Chromatic Number in time within a polynomial factor of $(2^{Delta+1-Delta-1)^{n/(Delta+1)$. For any constant $Delta$, these bounds are $O bigl((2-epsilon)^n bigr)$ for $epsilon>0$ independent of the number of vertices $n$.

BibTeX - Entry

@InProceedings{bjrklund_et_al:LIPIcs:2008:1336,
  author =	{Andreas Bj{\"o}rklund and Thore Husfeldt and Petteri Kaski and Mikko Koivisto},
  title =	{{Trimmed Moebius Inversion and Graphs of Bounded Degree}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{85--96},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1336},
  URN =		{urn:nbn:de:0030-drops-13369},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2008.1336},
  annote =	{Keywords: }
}

Seminar: 25th International Symposium on Theoretical Aspects of Computer Science
Issue date: 2008
Date of publication: 2008


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