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 $Gu0027$ and an integer $ku0027$ in $O(k^4 m)$ time such that (1) the size of the graph $Gu0027$ is $O(k^2)$, (2) $ku0027leq k$, and (3) $G$ has a feedback vertex set of size at most $k$ if and only if $Gu0027$ has a feedback vertex set of size at most $ku0027$. This is the first linear-time polynomial-size kernel for Feedback Vertex Set. The size of our kernel is $2k^2+k$ vertices and $4k^2$ edges, which is smaller than the previous best of $4k^2$ vertices and $8k^2$ edges. Thus, we improve the size and the running time simultaneously. note that under the assumption of $mathrm{NP}notsubseteqmathrm{coNP}/mathrm{poly}$, Feedback Vertex Set does not admit an $O(k^{2-epsilon})$-size kernel for any $epsilonu003e0$. 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. believe that this combinatorial algorithm is of independent interest. We have submitted a solver based on the proposed kernel and the efficient augmenting-path algorithm to the 1st parameterized algorithms and computational experiments challenge.