Papers › Linear-time Kernelization for Feedback Vertex Set
Linear-time Kernelization for Feedback Vertex Set
Yoichi Iwata
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In this paper, we propose an algorithm that, given an undirected graph G of m edges and an integer k, computes a graph G′ and an integer k′ in O(k⁴ m) time such that (1) the size of the graph G′ is O(k²), (2) k′≤k, and (3) G has a feedback vertex set of size at most k if and only if G′ has a feedback vertex set of size at most k′. This is the first linear-time polynomial-size kernel for Feedback Vertex Set. The size of our kernel is 2k²+k vertices and 4k² edges, which is smaller than the previous best of 4k² vertices and 8k² edges. Thus, we improve the size and the running time simultaneously. We note that under the assumption of NP⊈coNP/poly, Feedback Vertex Set does not admit an O(k^(2-ϵ))-size kernel for any ϵ>0. Our kernel exploits k-submodular relaxation, which is a recently developed technique for obtaining efficient FPT algorithms for various problems. The dual of k-submodular relaxation of Feedback Vertex Set can be seen as a half-integral variant of A-path packing, and to obtain the linear-time complexity, we propose an efficient augmenting-path algorithm for this problem. We believe that this combinatorial algorithm is of independent interest. A solver based on the proposed method won first place in the 1st Parameterized Algorithms and Computational Experiments (PACE) challenge.
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