Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

  • Rajesh Chitnis
  • Marek Cygan
  • Mohammadtaghi Hajiaghayi
  • Dániel Marx
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


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 4 k k!n O(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.


Undirected Graph Rected Graph Important Separator Undirected Version Iterative Compression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rajesh Chitnis
    • 1
  • Marek Cygan
    • 2
  • Mohammadtaghi Hajiaghayi
    • 1
  • Dániel Marx
    • 3
  1. 1.Department of Computer ScienceUniversity of Maryland at College ParkUSA
  2. 2.IDSIAUniversity of LuganoSwitzerland
  3. 3.Computer and Automation Research InstituteHungarian Academy of Sciences, (MTA SZTAKI)BudapestHungary

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