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. 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.