Faster fixed parameter tractable algorithms for finding feedback vertex sets

A feedback vertex set (<i>fvs</i>) of a graph is a set of vertices whose removal results in an acyclic graph. We show that if an undirected graph on <i>n</i> vertices with minimum degree at least 3 has a fvs on at most 1/3<i>n</i><sup>1 − ε</sup> vertices, then there is a cycle of length at most 6/ε (for ε ≥ 1/2, we can even improve this to just 6).Using this, we obtain a <i>O</i>((12 log <i>k</i>/log log <i>k</i> + 6)<sup>k</sup> <i>n</i><sup>ω</sup> algorithm for testing whether an undirected graph on <i>n</i> vertices has a fvs of size at most <i>k</i>. Here <i>n</i><sup>ω</sup> is the complexity of the best matrix multiplication algorithm. The previous best parameterized algorithm for this problem took <i>O</i>((2<i>k</i> + 1)<sup><i>k</i></sup><i>n</i><sup>2</sup>) time.We also investigate the fixed parameter complexity of weighted feedback vertex set problem in weighted undirected graphs.

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