Simpler and Better Approximation Algorithms for the Unweighted Minimum Label s-t Cut Problem

Abstract

Given a graph $$G=(V,E)$$ with a label set $$L = \{\ell _1, \ell _2, \ldots , \ell _q \}$$ , in which each edge has a label from L, and a source $$s \in V$$ together with a sink $$t \in V$$ , the Minimum Label s-t Cut problem asks to pick a set $$L' \subseteq L$$ of labels with minimized cardinality, such that the removal of all edges with labels in $$L'$$ from G disconnects s and t. Let $$n = |V|$$ and $$m = |E|$$ . The previous best known approximation ratio for this problem in literature is $$O(m^{1/2})$$ . We present two simple and purely combinatorial approximation algorithms for the problem with ratios $$O(n^{2/3}/\text {OPT}^{1/3})$$ and $$O(m^{1/2} / \text {OPT}^{1/2})$$ , where $$\text {OPT}$$ is the optimal value of the input instance. The former result gives the first approximation ratio which is sublinear in n for the problem, and in particular applies to the instances with dense graphs (e.g., $$m = \varTheta (n^2)$$ ). The latter result improves the previous ratio $$O(m^{1/2})$$ , as we can always assume that $$\text {OPT}$$ is a super-constant.