New algorithms for the Minimum Coloring Cut Problem

The Minimum Coloring Cut Problem is defined as follows: given a connected graph G with colored edges, find an edge cut E' of G (a minimal set of edges whose removal renders the graph disconnected) such that the number of colors used by the edges in E' is minimum. In this work, we present two approaches based on Variable Neighborhood Search to solve this problem. Our algorithms are able to find all the optimum solutions described in the literature

Solving the minimum labeling global cut problem by mathematical programming

Let G = (V, E, L) be an edge-labeled graph such that V is the set of vertices, E is the set of edges, L is the set of labels (colors) and each edge e \in E has a label l(e) associated; The goal of the minimum labeling global cut problem (MLGCP) is to find a subset L \subseteq L of labels such that G = (V, E , L\L ) is not connected and |L| is minimized. This work proposes three new mathematical formulations for the MLGCP as well as branch-and-cut algorithms to solve them. The computational experiments showed that the proposed methods are able to solve small to average sized instances in a reasonable amount of time

Minimum Label s-t Cut has Large Integrality Gaps

Given a graph G=(V,E) with a label set L = {l_1, l_2, ..., l_q}, in which each edge has a label from L, a source s in V, and a sink t in V, the Min 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. This problem comes from many applications in real world, for example, information security and computer networks. In this paper, we study two linear programs for Min Label s-t Cut, proving that both of them have large integrality gaps, namely, Omega(m) and Omega(m^{1/3-epsilon}) for the respective linear programs, where m is the number of edges in the graph and epsilon > 0 is any arbitrarily small constant. As Min Label s-t Cut is NP-hard and the linear programming technique is a main approach to design approximation algorithms, our results give negative answer to the hope that designs better approximation algorithms for Min Label s-t Cut that purely rely on linear programming