Fast Algorithms for Max Independent Set in Graphs of Small Average Degree

Details Fast Algorithms for Max Independent Set in Graphs of Small Average Degree Fast Algorithms for Max Independent Set in Graphs of Small Average Degree

2009-01-12

arxiv.org/abs/0901.1563v1

Computer Science

Discrete Mathematics

Scientific paper

Max Independent Set (MIS) is a paradigmatic problem in theoretical computer science and numerous studies tackle its resolution by exact algorithms with non-trivial worst-case complexity. The best such complexity is, to our knowledge, the $O^*(1.1889^n)$ algorithm claimed by J.M. Robson (T.R. 1251-01, LaBRI, Univ. Bordeaux I, 2001) in his unpublished technical report. We also quote the $O^*(1.2210^n)$ algorithm by Fomin and al. (in Proc. SODA'06, pages 18-25, 2006), that is the best published result about MIS. In this paper we settle MIS in (connected) graphs with "small" average degree, more precisely with average degree at most 3, 4, 5 and 6. Dealing with graphs of average degree at most 3, the best bound known is the recent $O^*(1.0977^n)$ bound by N. Bourgeois and al. in Proc. IWPEC'08, pages 55-65, 2008). Here we improve this result down to $O^*(1.0854^n)$ by proposing finer and more powerful reduction rules. We then propose a generic method showing how improvement of the worst-case complexity for MIS in graphs of average degree $d$ entails improvement of it in any graph of average degree greater than $d$ and, based upon it, we tackle MIS in graphs of average degree 4, 5 and 6. For MIS in graphs with average degree 4, we provide an upper complexity bound of $O^*(1.1571^n)$ that outperforms the best known bound of $O^*(1.1713^n)$ by R. Beigel (Proc. SODA'99, pages 856-857, 1999). For MIS in graphs of average degree at most 5 and 6, we provide bounds of $O^*(1.1969^n)$ and $O^*(1.2149^n)$, respectively, that improve upon the corresponding bounds of $O^*(1.2023^n)$ and $O^*(1.2172^n)$ in graphs of maximum degree 5 and 6 by (Fomin et al., 2006).

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