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DOI: 10.4230/LIPIcs.STACS.2008.1336
URN: urn:nbn:de:0030-drops-13369
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2008/1336/
Björklund, Andreas ;
Husfeldt, Thore ;
Kaski, Petteri ;
Koivisto, Mikko
Trimmed Moebius Inversion and Graphs of Bounded Degree
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 = {10.4230/LIPIcs.STACS.2008.1336},
annote = {Keywords: }
}
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Collection: |
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25th International Symposium on Theoretical Aspects of Computer Science |
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Issue Date: |
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2008 |
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Date of publication: |
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06.02.2008 |
