Approximation algorithms for feasible cut and multicut problems

Abstract

Let G = (V, E) be an undirected graph with a capacity function u: E→R+ and let S1, S2,…, Sk be k commodities, where each Siconsists of a pair of nodes. A set S of nodes is called feasible if it contains no Si, 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 In 2)-approximation algorithm for the minimum capacity feasible v*-cut problem. The multicut problem is to find a set of edges F ⊆ E of minimum capacity such that no connected component of G \ F contains a commodity Si. We show that an a-approximation algorithm for the minimum-ratio feasible cut problem gives a 2a(1 + In T)-approximation algorithm for the multicut problem, where T denotes the cardinality of Ui Si. We give a new approximation guarantee of O(t log T) for the minimum capacity-to-demand ratio Steiner cut problem; here each Si is a set of nodes and t denotes the maximum cardlnality of a commodity Si.