# Solving the minimum labeling global cut problem by mathematical programming

## TL;DR

This paper introduces three new mathematical formulations and branch-and-cut algorithms to solve the minimum labeling global cut problem in edge-labeled graphs, demonstrating effectiveness on small to medium instances.

## Contribution

The work presents novel mathematical models and algorithms for the MLGCP, advancing solution methods for this combinatorial optimization problem.

## Key findings

- Proposed formulations effectively solve small to medium instances.
- Branch-and-cut algorithms outperform existing methods on tested instances.
- The methods provide a practical approach to the MLGCP.

## Abstract

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.