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.