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 S ₁, S ₂,⋯, S _k be k commodities, where each S_i 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 .

Supported in part by NSERC grant no. OGP0138432 (NSERC code OGPIN 007).