A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem
SIAM Journal on Discrete Mathematics
An 8-Approximation Algorithm for the Subset Feedback Vertex Set Problem
SIAM Journal on Computing
Approximating Minimum Feedback Sets and Multi-Cuts in Directed Graphs
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
WG '91 Proceedings of the 17th International Workshop
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
A fixed-parameter algorithm for the directed feedback vertex set problem
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Improved algorithms for feedback vertex set problems
Journal of Computer and System Sciences
Randomized algorithms for the loop cutset problem
Journal of Artificial Intelligence Research
Fixed-parameter tractability of multicut parameterized by the size of the cutset
Proceedings of the forty-third annual ACM symposium on Theory of computing
Subset feedback vertex set is fixed-parameter tractable
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Feedback vertex set in mixed graphs
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Compression via matroids: a randomized polynomial kernel for odd cycle transversal
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On feedback vertex set new measure and new structures
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Finding odd cycle transversals
Operations Research Letters
Parameterized Complexity
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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. Bodlaender (WG '91) gave the first fixed-parameter algorithm for FVS in undirected graphs. The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is fixed-parameter tractable by giving an 4kk!nO(1) algorithm. In the 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). Indeed both the edge and vertex versions are known to be equivalent in the parameterized sense. Recently the Subset Feedback Vertex Set in undirected graphs was shown to be FPT by Cygan et al. (ICALP '11) and Kakimura et al. (SODA '12). We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time $2^{2^{O(k)}}n^{O(1)}$, i.e., FPT parameterized by size k of the solution. By our result, we complete the picture for feedback vertex set 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) to show that Undirected Multicut is FPT and was generalized by Chitnis et al. (SODA '12) to directed graphs to show that Directed Multiway Cut is FPT. In this paper we give a general family of problems (which includes Directed Multiway Cut and Directed Subset Feedback Vertex Set among others) for which we can do random sampling of important separators and obtain a set which is disjoint from a minimum solution and covers its "shadow". We believe this general approach will be useful for showing the fixed-parameter tractability of other problems in directed graphs.