235 research outputs found
Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
Given a directed graph , a set of terminals and an integer , the
\textsc{Directed Vertex Multiway Cut} problem asks if there is a set of at
most (nonterminal) vertices whose removal disconnects each terminal from
all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous
problem where is a set of at most edges. These two problems indeed are
known to be equivalent. A natural generalization of the multiway cut is the
\emph{multicut} problem, in which we want to disconnect only a set of given
pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in
undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized
by . Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and
directed multicut is W[1]-hard parameterized by . We complete the picture
here by our main result which is that both \textsc{Directed Vertex Multiway
Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time
, i.e., FPT parameterized by size of the cutset of
the solution. This answers an open question raised by Marx (Theor. Comp. Sci.
2006) and Marx and Razgon (STOC 2011). It follows from our result that
\textsc{Directed Multicut} is FPT for the case of terminal pairs, which
answers another open problem raised in Marx and Razgon (STOC 2011)
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