Approximating the Weighted Minimum Label s-t Cut Problem
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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.
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