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The Complexity of Multiterminal Cuts
 SIAM Journal on Computing
, 1994
"... In the Multiterminal Cut problem we are given an edgeweighted 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 mincut, maxflow problem, and ..."
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In the Multiterminal Cut problem we are given an edgeweighted 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 mincut, maxflow problem, and can be solved in polynomial time. We show that the problem becomes NPhard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NPhard, 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.
Optimal 3Terminal Cuts and Linear Programming
"... . Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3cut is a subset A of E such that no two terminals are in the same component of GnA. If a nonnegative edge weight ce is specified for each e 2 E, the optimal 3cut problem is to find a 3cut of minimum total ..."
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. Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3cut is a subset A of E such that no two terminals are in the same component of GnA. If a nonnegative edge weight ce is specified for each e 2 E, the optimal 3cut problem is to find a 3cut of minimum total weight. This problem is NPhard, and in fact, is maxSNPhard. An approximation algorithm having performance guarantee 7 6 has recently been given by Calinescu, Karloff, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3cut has weight at most 7 6 times the optimal LP value. It is proved here that 7 6 can be improved to 12 11 , and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3cut problem having performance guarantee 12 11 . 1 Introduction Given an undirected graph G = (V; E) and k specified terminal nodes t 1 ; : : : ; t k , a kcut is a subset A of E such that no two term...