Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset

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).

💡 Research Summary

The paper addresses the long‑standing open problem of whether the directed multiway cut problem is fixed‑parameter tractable (FPT) when parameterized by the size $p$ of the cutset. In the directed vertex version, we are given a directed graph $G$, a set $T$ of $k$ terminals, and an integer $p$, and we must decide whether there exists a set $S$ of at most $p$ non‑terminal vertices whose removal destroys all directed paths between any two distinct terminals. The edge version, where $S$ consists of edges, is shown to be polynomial‑time equivalent to the vertex version.

The authors present an algorithm that solves both versions in time $2^{2^{O(p)}}\cdot n^{O(1)}$, thereby establishing that the problem is FPT with respect to $p$. This resolves the question raised by Marx (2006) and by Marx and Razgon (STOC 2011). As a corollary, the directed multicut problem with exactly two terminal pairs ($k=2$) also becomes FPT, answering another open problem from the same STOC paper.

Technical Overview
The algorithm builds on two central concepts that have become standard in parameterized cut problems: important separators and shadow removal. An important separator for a pair of vertex sets $(A,B)$ is a minimal $A$–$B$ cut that is inclusion‑wise minimal among all cuts of the same size; crucially, for a fixed bound $p$, the number of important $p$‑separators is bounded by $2^{O(p)}$. The authors enumerate all important $p$‑separators for every ordered terminal pair $(t_i,t_j)$, obtaining a global candidate set $C$ of size $2^{O(p)}$.

The second ingredient, shadow removal, deals with the “shadow” – the region of the graph that becomes unreachable from the terminals after deleting a hypothetical solution $S$, but whose vertices are not explicitly part of $S$. The paper introduces a randomized sampling technique that, with high probability, selects a small set $R$ intersecting the shadow in a controlled way. By fixing $R$, the problem reduces to finding a cut inside $C\cup R$ of size at most $p$, which can be solved by standard flow‑based methods because the candidate set is now of bounded size.

To glue these components together, the authors employ iterative compression. Starting from a trivial solution of size $p+1$, they repeatedly compress a solution of size $p+1$ to one of size $p$ using the above separator enumeration and shadow‑removal subroutine. Each compression step runs in $2^{O(p)}$ time, and the number of steps is linear in $n$, leading to the overall $2^{2^{O(p)}}\cdot n^{O(1)}$ bound.

Complexity Landscape
Prior work established that undirected multiway cut is FPT (Marx, 2006) and that undirected multicut is also FPT, while directed multicut is W