The complexity of multiway cuts (extended abstract)

Authors:
E. Dahlhaus;D. S. Johnson;C. H. Papadimitriou;P. D. Seymour;M. Yannakakis
Affiliations:
Department of Computer Science and University of Sidney, New South Wales, Australia and University of Bonn;AT&T Bell Laboratories, Murray Hill, NJ;Department of Computer Science and Engineering, University of California at San Diego;Bellcore, Morristown, NJ;AT&T Bell Laboratories, Murray Hill, NJ
Venue:
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Year:
1992

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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.