Faster Fixed Parameter Tractable Algorithms for Undirected Feedback Vertex Set
We give a O(maxû12k, (4lgk)ký · nω) algorithm for testing whether an undirected graph on n vertices has a feedback vertex set of size at most k where O(nω) is the complexity of the best matrix multiplication algorithm. The previous best fixed parameter tractable algorithm for the problem took O((2k + 1)k n2) time. The main technical lemma we prove and use to develop our algorithm is that that there exists a constant c such that, if an undirected graph on n vertices with minimum degree 3 has a feedback vertex set of size at most c√n, then the graph will have a cycle of length at most 12. This lemma may be of independent interest.
We also show that the feedback vertex set problem can be solved in O(dkkn) for some constant d in regular graphs, almost regular graphs and (fixed) bounded degree graphs.
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