Approximation algorithms for feasible cut and multicut problems

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Let G=(V, E) be an undirected graph with a capacity function u:E→ℜ + and let S 1, S 2,⋯, S k be k commodities, where each Si consists of a pair of nodes. A set S of nodes is called feasible if it contains no S i , and a cut (S, ¯S) is called feasible if S is feasible. We show that several optimization problems on feasible cuts are NP-hard. We give a (4 ln 2)-approximation algorithm for the minimum capacity feasible v*-cut problem. The multicut problem is to find a set of edges F \(\subseteq\) E of minimum capacity such that no connected component of G/F contains a commodity S i . We show that an α-approximation algorithm for the minimum-ratio feasible cut problem gives a 2α(1+ln T)-approximation algorithm for the multicut problem, where T denotes the cardinality of \(\cup _i S_i\). We give a new approximation guarantee of O(t log T) for the minimum capacity-to-demand ratio Steiner cut problem; here each S i is a set of nodes and t denotes the maximum cardinality of a commodity S i .