Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Given a graph G and an integer k, the Feedback Vertex Set (FVS) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph. The first fixed-parameter algorithm for FVS in undirected graphs appeared in a monograph of Mehlhorn in 1984. The fixed-parameter tractability (FPT) status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC ’08, JACM ’08) showed that it is fixed-parameter tractable by giving a 4^kk! · n^O(1) time algorithm. There are two subset versions of this problems: We are given an additional subset S of vertices (resp., edges), and we want to hit all cycles passing through a vertex of S (resp., an edge of S); the two variants are known to be equivalent in the parameterized sense. Recently, the Subset FVS problem in undirected graphs was shown to be FPT by Cygan et al. (ICALP’11, SIDMA’13) and independently by Kakimura et al. (SODA ’12). We generalize the result of Chen et al. (STOC ’08, JACM ’08) by showing that a Subset FVS in directed graphs can be solved in time 2^O(k3)ċn^O(1) (i.e., FPT parameterized by size k of the solution). By our result, we complete the picture for FVS problems and their subset versions in undirected and directed graphs. The technique of random sampling of important separators was used by Marx and Razgon (STOC ’11, SICOMP ’14) to show that Undirected Multicut is FPT, and it was generalized by Chitnis et al. (SODA ’12, SICOMP ’13) to directed graphs to show that Directed Multiway Cut is FPT. In addition to proving the FPT of a Directed Subset FVS, we reformulate the random sampling of important separators technique in an abstract way that can be used with a general family of transversal problems. We believe this general approach will be useful for showing the FPT of other problems in directed graphs. Moreover, we modify the probability distribution used in the technique to achieve better running time; in particular, this gives an improvement from 2^2O(k) to 2^O(k2) in the parameter dependence of the Directed Multiway Cut algorithm of Chitnis et al. (SODA ’12, SICOMP ’13).