Optimal 3-Terminal Cuts and Linear Programming

Given an undirected graph G = (V; E) and three speciied terminal nodes t1; t2; t3, a 3-cut is a subset A of E such that no two terminals are in the same component of GnA. If a non-negative edge weight ce is speciied for each e 2 E, the optimal 3-cut problem is to nd a 3-cut of minimum total weight. This problem is NP-hard, and in fact, is max-SNP-hard. An approximation algorithm having performance guarantee 7 6 has recently been given by CC alinescu, Karloo, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3-cut 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 3-cut problem having performance guarantee 12 11 .