In this paper, we study labeled extensions of the classical s,t-mincut problem, in which we are given a graph G=(V,E), two specific vertices s,t∈V, a set L of labels, and a labeling ℓ:E→L of the edges. The goal is to choose a subset L′⊆L of labels, so that s and t become disconnected when deleting the edges with labels in L′. We give an algorithm with an O(n2/3) approximation factor guarantee, which improves the O(m) approximation guarantee of Zhang et al. (2009) . We also consider variants in which selected subsets of paths between s and t have to be removed (instead of all paths). These labeled cut problems are much harder than the classical mincut problem.
Dutta, T., Heath, L. S., Kumar, V. S. A., & Marathe, M. V. (2016). Labeled cuts in graphs. Theoretical Computer Science, 648, 34–39. https://doi.org/10.1016/j.tcs.2016.07.040