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# Trimmed Moebius Inversion and Graphs of Bounded Degree

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LIPIcs.STACS.2008.1336.pdf
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## Cite As

Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Trimmed Moebius Inversion and Graphs of Bounded Degree. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 85-96, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)
https://doi.org/10.4230/LIPIcs.STACS.2008.1336

## 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\$.

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