Given a directed graph $G$, a set of $k$ terminals and an integer $p$, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where $S$ is a set of at most $p$ 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 $k$ 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 $p$. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]-hard parameterized by $p$. 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 $2^{2^{O(p)}}n^{O(1)}$, i.e., FPT parameterized by size $p$ 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 $k=2$ terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011).
Ford and Fulkerson [11] gave the classical result on finding a minimum cut that separates two terminals s and t in 1956. A natural and well-studied generalization of the minimum s -t cut problem is Multiway Cut, in which given a graph G and a set of terminals {s 1 , s 2 , . . . , s k }, the task is to find a minimum subset of vertices or edges whose deletion disconnects all the terminals from one another. Dahlhaus et al. [8] showed the edge version in undirected graphs is APX-complete for k ≥ 3. For the edge version Karger et al. [15] gave the current best known approximation ratio of 1.3438 for general k. The vertex version of the problem is known to be at least as hard as the edge version, and the current best approximation ratio is 2 -2 k [13]. The problem behaves very differently on directed graphs. Interestingly, for directed graphs, the edge and vertex versions turn out to be equivalent. Garg et al. [13] showed that computing a minimum multiway cut in directed graphs is NP-hard and MAX SNP-hard already for k = 2.
Paper Vertex Version Non-constructive FPT Roberston and Seymour [24,25] O * (4 p 3 ) Marx [ They also give an approximation algorithm with ratio 2 log k, which was improved to ratio 2 later by Naor and Zosin [21].
Rather than finding approximate solutions in polynomial time, one can look for exact solutions in time that is superpolynomial, but still better than the running time obtained by brute force solutions. For example, Dahlhaus et al. [8] showed that undirected Multiway Cut can be solved in time n O(k) on planar graphs, which can be an efficient solution if the number of terminals is small. On the other hand, on general graphs the problem becomes NP-hard already for k = 3. In both the directed and the undirected version, brute force can be used to check in time n O(p) if a solution of size at most p exists: one can go through all sets of size at most p. Thus the problem can be solved in polynomial time if the optimum is assumed to be small. In the undirected case, significantly better running time can be obtained: the current fastest algorithms run in O * (2 p ) time for both the vertex version [7] and the edge version [26] (the O * notation hides all factors which are polynomial in size of input). That is, undirected Multiway Cut is fixed-parameter tractable parameterized by the size of the cutset we remove. Recall that a problem is fixed-parameter tractable (FPT) with a particular parameter p if it can be solved in time f (p)n O (1) , where f is an arbitrary function depending only on p; see [9,10,22] for more background. We give a brief summary of the race for faster FPT algorithms for Undirected Multiway Cut in Figure 1.
Our main result is that the directed version of Multiway Cut is also fixed-parameter tractable: Note that the hardness result of Garg et al. [13] shows that in the directed case the problem is nontrivial (in fact, NP-hard) even for k = 2 terminals; our result holds without any bound on the number of terminals. The question was first asked explicitly in [18] and was also stated as an open problem in [19]. Our result shows in particular that directed multiway cut is solvable in polynomial time if the size of the optimum solution is O(log log n), where n is the number of vertices in the digraph.
A more general problem is Multicut: the input contains a set {(s 1 , t 1 ), . . . , (s k , t k )} of k pairs, and the task is to break every path from s i to its corresponding t i by the removal of at most p vertices. Very recently, it was shown that undirected Multicut is FPT parameterized by p [1,19], but the directed version is unlikely to be FPT as it is W [1]-hard [19] with this parameterization. However, in the special case of k = 2 terminal pairs, there is a simple reduction from Directed Multicut to Directed Multiway Cut, thus our result shows that the latter problem is FPT parameterized by p for k = 2. Let us briefly sketch the reduction. (Note that the reduction we sketch works only for the variant of Directed Multicut which allows the deletion of terminals.
Marx and Razgon [19] asked about the FPT status of this variant which is in fact equivalent to the one which does not allow deletion of the terminals.) Let (G, T, p) be a given instance of Directed Multicut and let T = {(s 1 , t 1 ), (s 2 , t 2 )}. We construct an equivalent instance of Directed Multiway Cut as follows: Graph G is obtained by adding two new vertices s, t to the graph and adding the four edges s → s 1 , t 1 → t, t → s 2 , and t 2 → s. It is easy to see that the Directed Multiway Cut instance (G , {s, t}, p) is equivalent to the original Directed Multicut instance.
The complexity of the case k = 3 remains an interesting open problem.
Our techniques. Our algorithm for Directed Multiway Cut is inspired by the algorithm of Marx and Razgon [19] for undirected Multicut. In particular we use the technique of “random sampling of important separators” introduced in [19] and try to ensure that there
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