Parameterized Algorithms for Feedback Vertex Set

We present an algorithm for the parameterized feedback vertex set problem that runs in time \(O((2\lg{k}+ 2\lg{\lg{k}+ 18})^k n^2)\). This improves the previous \(O(max\{12^k, (4\lg{k})^k\}n^{\omega})\) algorithm by Raman et al. by roughly a 2 k factor (n w ∈ O(n 2.376) is the time needed to multiply two n × n matrices). Our results are obtained by developing new combinatorial tools and employing results from extremal graph theory. We also show that for several special classes of graphs the feedback vertex set problem can be solved in time c k n O(1) for some constant c. This includes, for example, graphs of genus \(O(\lg{n})\).

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