The complexity of multiway cuts (extended 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 <italic>k</italic> 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 <italic>k</italic> = 3, but can be solved in polynomial time for planar graphs for any fixed <italic>k</italic>. The planar problem is NP-hard, however, if <italic>k</italic> is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2–2/<italic>k</italic> of the optimal cut weight.

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