Approximating Minimum Label s-t Cut via Linear Programming

Abstract

We consider the Minimum Label s-t Cut problem. Given an undirected graph G = (V,E) with a label set L, in which each edge has a label from L, and a source s ∈ V together with a sink t ∈ V, the goal of the Minimum Label s-t Cut problem is to pick a subset of labels of minimized cardinality, such that the removal of all edges with these labels from G disconnects s and t. We present a min { O((m/OPT)^1/2), O(n ^2/3/OPT ^1/3) }-approximation algorithm for the Minimum Label s-t Cut problem using linear programming technique, where n = |V|, m = |E|, and OPT is the optimal value of the input instance. This result improves the previously best known approximation ratio O(m ^1/2) for this problem (Zhang et al., JOCO 21(2), 192–208 (2011)), and gives the first approximation ratio for this problem in terms of n. Moreover, we show that our linear program relaxation for the Minimum Label s-t Cut problem, even in a stronger form, has integrality gap Ω((m/OPT)^{1/2 − ε}).