The Minimum k-way Cut of Bounded Size is Fixed-Parameter Tractable

We consider the minimum k-way cut problem for unweighted undirected graphs with a size bound s on the number of cut edges allowed. Thus we seek to remove as few edges as possible so as to split a graph into k components, or report that this requires cutting more than s edges. We show that this problem is fixed-parameter tractable (FPT) with the standard parameterization in terms of the solution size s. More precisely, for s=O(1), we present a quadratic time algorithm. Moreover, we present a much easier linear time algorithm for planar graphs and bounded genus graphs. Our tractability result stands in contrast to known W[1] hardness of related problems. Without the size bound, Downey et al. [2003] proved that the minimum k-way cut problem is W[1] hard with parameter k, and this is even for simple unweighted graphs. Downey et al. asked about the status for planar graphs. We get linear time with fixed parameter k for simple planar graphs since the minimum k-way cut of a planar graph is of size at most 6k. More generally, we get FPT with parameter k for any graph class with bounded average degree. A simple reduction shows that vertex cuts are at least as hard as edge cuts, so the minimum k-way vertex cut is also W[1] hard with parameter k. Marx [2004] proved that finding a minimum k-way vertex cut of size s is also W[1] hard with parameter s. Marx asked about the FPT status with edge cuts, which we prove tractable here. We are not aware of any other cut problem where the vertex version is W[1] hard but the edge version is FPT, e.g., Marx [2004] proved that the k-terminal cut problem is FPT parameterized by the cut size, both for edge and vertex cuts.