On feedback vertex set new measure and new structures

Authors:
Yixin Cao;Jianer Chen;Yang Liu
Affiliations:
Department of Computer Science and Engineering, Texas A&M University;Department of Computer Science and Engineering, Texas A&M University;Department of Computer Science, University of Texas - Pan American
Venue:
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Year:
2010

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Abstract

We study the parameterized complexity of the feedback vertex set problem (fvs) on undirected graphs. We approach the problem by considering a variation of it, the disjoint feedback vertex set problem (disjoint-fvs), which finds a disjoint feedback vertex set of size k when a feedback vertex set of a graph is given. We show that disjoint-fvs admits a small kernel, and can be solved in polynomial time when the graph has a special structure that is closely related to the maximum genus of the graph. We then propose a simple branch-and-search process on disjoint-fvs, and introduce a new branch-and-search measure. The branch-and-search process effectively reduces a given graph to a graph with the special structure, and the new measure more precisely evaluates the efficiency of the branch-and-search process. These algorithmic, combinatorial, and topological structural studies enable us to develop an O(3.83kkn2) time parameterized algorithm for the general fvs problem, improving the previous best algorithm of time O(5kkn2) for the problem.