Approximating the Weighted Minimum Label $s$-$t$ Cut Problem

Abstract

In the weighted (minimum) Label s-t Cut problem, we are given a (directed or undirected) graph G = (V, E), a label set L = {ℓ 1 , ℓ 2 , . . . , ℓ q } with positive label weights {w ℓ }, a source s ∈ V and a sink t ∈ V . Each edge edge e of G has a label ℓ(e) from L. Different edges may have the same label. The problem asks to find a minimum weight label subset L ′ such that the removal of all edges with labels in L ′ disconnects s and t.The unweighted Label s-t Cut problem (i.e., every label has a unit weight) can be approximated within O(n 2/3 ), where n is the number of vertices of graph G. However, it is unknown for a long time how to approximate the weighted Label s-t Cut problem within o(n). In this paper, we provide an approximation algorithm for the weighted Label s-t Cut problem with ratio O(n 2/3 ). The key point of the algorithm is a mechanism to interpret label weight on an edge as both its length and capacity.