An exact algorithm for maximum independent set in degree-5 graphs

The maximum independent set problem is a basic NP-hard problem and has been extensively studied in exact algorithms. The maximum independent set problems in low-degree graphs are also important and may be bottlenecks of the problem in general graphs. In this paper, we present an O∗(1.1737n)-time exact algorithm for the maximum independent set problem in an n-vertex graph with degree bounded by 5, improving the previous running time bound of O∗(1.1895n). In our algorithm, we introduce an effective divide-and-conquer procedure to deal with vertex cuts of size at most two in graphs, and design branching rules on some special structure of triconnected graphs of maximum degree 5. These result in an improved algorithm without introducing a large number of branching rules.

[1]  Robert E. Tarjan,et al.  Finding a Maximum Independent Set , 1976, SIAM J. Comput..

[2]  David Eppstein,et al.  Quasiconvex analysis of backtracking algorithms , 2003, SODA '04.

[3]  Tang Jian,et al.  An O(20.304n) Algorithm for Solving Maximum Independent Set Problem , 1986, IEEE Trans. Computers.

[4]  Fabrizio Grandoni,et al.  A measure & conquer approach for the analysis of exact algorithms , 2009, JACM.

[5]  Weijia Jia,et al.  Improvement on vertex cover for low-degree graphs , 2000, Networks.

[6]  John Michael Robson,et al.  Algorithms for Maximum Independent Sets , 1986, J. Algorithms.

[7]  Hiroshi Nagamochi,et al.  Confining sets and avoiding bottleneck cases: A simple maximum independent set algorithm in degree-3 graphs , 2013, Theor. Comput. Sci..

[8]  Igor Razgon.  Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3 , 2009, J. Discrete Algorithms.

[9]  D. West.  Introduction to Graph Theory , 1995 .

[10]  Ge Xia,et al.  Improved upper bounds for vertex cover , 2010, Theor. Comput. Sci..

[11]  Vangelis Th. Paschos,et al.  Fast Algorithms for max independent set , 2010, Algorithmica.

[12]  Vangelis Th. Paschos,et al.  Maximum Independent Set in Graphs of Average Degree at Most Three in O(1.08537n){\mathcal O}(1.08537^n) , 2010, TAMC.

[13]  Fedor V. Fomin,et al.  Reports in Informatics , 2005 .

[14]  Mingyu Xiao.  A Simple and Fast Algorithm for Maximum Independent Set in 3-Degree Graphs , 2010, WALCOM.

[15]  Martin Fürer,et al.  A Faster Algorithm for Finding Maximum Independent Sets in Sparse Graphs , 2006, LATIN.

[16]  Joachim Kneis,et al.  A Fine-grained Analysis of a Simple Independent Set Algorithm , 2009, FSTTCS.