An O∗(2O(k)) FPT Algorithm for the Undirected Feedback Vertex Set Problem∗
نویسندگان
چکیده
We describe an algorithm for the FEEDBACK VERTEX SET problem on undirected graphs, parameterized by the size k of the feedback vertex set, that runs in time O(ckn3) where c = 10.567 and n is the number of vertices in the graph. This improves on the best previous FPT algorithms due to [RSS02, KPS04] and [RS05] that have running times of the form O ∗(2O(k lg lg ) and O∗(2O(k lg k lg lg k ), respectively. These previous algorithms were based on the method of bounded search trees, branching on short cycles. Our algorithm is based on the relatively new FPT technique of iterative compression, and we prove our main result for a more general “annotated” form of the problem, where a subset of the vertices may be marked as not to belong to the feedback vertex set. We also establish “exponential optimality” for our algorithm; we prove that no FPT algorithm with a running time of the form O ∗(2o(k)) is possible, unless there is an unlikely collapse of parameterized complexity classes, FPT = M [1].
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