An exact algorithm for the adjacent vertex distinguishing sum edge coloring problem

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics Pub Date : 2025-03-29 DOI:10.1016/j.dam.2025.03.029

Brian Curcio, Isabel Méndez-Díaz, Paula Zabala

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引用次数: 0

Abstract

In this work we define the adjacent vertex distinguishing sum edge coloring problem. This problem consists of finding an assignment of colors to the edges of a graph with the following constraints: every pair of adjacent edges must have a different color, and every pair of adjacent vertices must not have the same set of colors assigned to the edges incident to each. The goal is to minimize the sum of the colors in an edge coloring that satisfies these constraints. This problem is a special case of a large family of problems known as graph labeling, which is a widely used and very popular set of tools to build abstract models for problems that arise in everyday life.

Some variants of graph labeling problems have been successfully addressed with mixed-integer linear programming (MIP) techniques based on a polyhedral characterization of the set of feasible solutions. We use this approach to develop a Branch and Cut algorithm to solve the problem.

We propose two MIP models that are computationally evaluated to choose the most promising one and continue with a polyhedral study. This analysis aims to characterize valid inequalities that strengthen the formulation in the hope of improving the algorithm’s performance. These inequalities are added on demand as cutting planes using exact and heuristic separation algorithms. Additionally, we considered the use of an initial heuristic and a specific branching strategy.

The results show that the algorithm developed allows us to solve instances that were unsolvable using general-purpose solvers. Our polyhedral study and the addition of cutting planes have proved to be crucial factors in solving the most challenging instances.

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来源期刊

Discrete Applied Mathematics 数学-应用数学

CiteScore

2.30

自引率

9.10%

发文量

422

审稿时长

4.5 months

期刊介绍： The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal. Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.