Linear-Time Kernelization for Feedback Vertex Set

In this paper, we give 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(km) time such that (1) the size of the graph G′ is O(k), (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 2k + k vertices and 4k edges, which is smaller than the previous best of 4k vertices and 8k edges. Thus, we improve the size and the running time simultaneously. We note that under the assumption of NP 6⊆ coNP/poly, Feedback Vertex Set does not admit an O(k2− )-size kernel for any > 0. Our kernel exploits k-submodular relaxation, which is a recently developed technique for obtaining efficient FPT algorithms for various problems. The 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 give an efficient augmenting-path algorithm for this problem. We believe that this combinatorial algorithm is of independent interest. A solver based on the proposed kernel and the efficient augmenting-path algorithm has been submitted to the 1st parameterized algorithms and computational experiments challenge and won the first place.