The complexity of multiway cuts (extended abstract)
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TL;DR
It is shown that the Multiway Cut problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k, and a simple approximation algorithm is described that is guaranteed to come within a factor of 2–2/k of the optimal cut weight.
Abstract
In the Multiway Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the min-cut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2–2/k of the optimal cut weight.