The Complexity of Multiterminal Cuts

In the multiterminal cut problem one is given an edge-weighted graph and a subset of the vertices called terminals, and is 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 mincut, max-flow problem, and can be solved in polynomial time. It is shown 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. A simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of $2-2/k$ of the optimal cut weight is also described.

[1]  S. Vajda,et al.  Integer Programming and Network Flows , 1970 .

[2]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[3]  Harold S. Stone,et al.  Multiprocessor Scheduling with the Aid of Network Flow Algorithms , 1977, IEEE Transactions on Software Engineering.

[4]  R. E. Bellman,et al.  Review: Eugene L. Lawler, Combinatorial optimization: networks and matroids , 1978 .

[5]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[7]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[8]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[9]  Mihalis Yannakakis,et al.  Cutting and Partitioning a Graph aifter a Fixed Pattern (Extended Abstract) , 1983, ICALP.

[10]  D. Hochbaum,et al.  An $O ( | V |^2 )$ Algorithm for the Planar 3-Cut Problem , 1985 .

[11]  Greg N. Frederickson,et al.  Fast Algorithms for Shortest Paths in Planar Graphs, with Applications , 1987, SIAM J. Comput..

[12]  Dorit S. Hochbaum,et al.  Polynomial algorithm for the k-cut problem , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[13]  A. Goldberg,et al.  A new approach to the maximum-flow problem , 1988, JACM.

[14]  William H. Cunningham.  The Optimal Multiterminal Cut Problem , 1989, Reliability Of Computer And Communication Networks.

[15]  M. R. Rao,et al.  On the multiway cut polyhedron , 1991, Networks.

[16]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[17]  Vijay V. Vazirani,et al.  Finding k-cuts within twice the optimal , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[18]  Mihalis Yannakakis,et al.  The complexity of multiway cuts (extended abstract) , 1992, STOC '92.

[19]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[20]  László A. Székely,et al.  Algorithms and Min-max Theorems for Certain Multiway Cuts , 1992, IPCO.

[21]  P. Erdös,et al.  Evolutionary trees: An integer multicommodity max-flow-min-cut theorem , 1992 .

[22]  Mihalis Yannakakis,et al.  Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications , 1996, SIAM J. Comput..