求平均度小的图的最大独立集的快速算法
Fast Algorithms for Max Independent Set in Graphs of Small Average Degree
摘要：
最大独立集（MIS）问题是理论计算机科学中的一个典型问题，许多研究都是通过具有非平凡最坏情况复杂度的精确算法来解决。据我们所知，最好的这种复杂度是由J.M.罗布森（T.R. 1251-01，LaBRI，波尔多I大学，2001）。我们还引用了Fomin等人的$O^*(1.2210^n)$算法（Proc. SODA'06，pages 18-25，2006），这是关于MIS的最佳公开结果。在本文中，我们解决MIS的（连通）图与“小”的平均度，更准确地说，平均度最多为3，4，5和6。对于平均度不超过3的图，已知的最佳界是最近的$O^*(1.0977^n)$被N界。Bourgeois等人，Proc. IWPEC'08，第55-65页，2008）。在这里，我们通过提出更精细和更强大的约简规则将此结果改进为$O^*(1.0854^n)$。然后，我们提出了一个通用的方法，显示如何改善MIS的最坏情况下的复杂性的平均度$d$的图需要改善它在任何图的平均度大于$d$，并在此基础上，我们解决MIS的平均度4，5和6的图。对于平均度为4的图中的MIS，我们给出了$O^*(1.1713^n)$的一个复杂性上界，该上界优于R. Beigel（Proc. SODA'99，第856-857页，1999）。对于平均度至多为5和6的图中的MIS，我们分别提供了$O^*(1.1969^n)$和$O^*(1.2149^n)$的界，其改进了最大度为5和6的图中的$O^*(1.2023^n)$和$O^*(1.2172^n)$的相应界（Fomin等人，2006年）。
英文摘要：
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 Bordeaux I, 2001) in his unpublished technical report.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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