Applying Genetic Algorithms to Validate a Conjecture in Graph Theory: The Minimum Dominating Set Problem

Authors

  • Jorge Cervantes-Ojeda Universidad Autónoma Metropolitana
  • María C. Gómez-Fuentes Universidad Autónoma Metropolitana
  • Julian A. Fresán-Figueroa Universidad Autónoma Metropolitana

DOI:

https://doi.org/10.13053/cys-27-4-4782

Keywords:

GA for theoretical problems, rank GA, genetic algorithms, minimum dominating set

Abstract

This paper presents a case study where the interdisciplinary approach between artificial intelligence, specifically genetic algorithms, and discrete mathematics has been instrumental in verifying a conjecture in graph theory. The focus of our research lies in the Minimum Dominating Set (MDS) problem, which involves identifying a dominating set with the minimum cardinality for a given graph. While previous works have primarily aimed at utilizing genetic algorithms to efficiently find satisfactory solutions to the MDS problem, our study aims to discover the optimal solution. The Rank GA algorithm employed in our work possesses the ability to escape local optima while simultaneously conducting local search to refine the best available solution. Since the MDS problem is known to be NP-hard, the characteristics of Rank GA, coupled with the identification of regularities in one of the solutions, enabled us to verify the conjecture for graphs comprising 30, 80, 312, and 800 vertices. This research serves as a testament to the versatility of genetic algorithms, showcasing their utility not only in practical problem-solving but also in tackling theoretical challenges.

Downloads

Published

2023-12-17

Issue

Vol. 27 No. 4 (2023): 27(4) 2023

Section

Articles

License 

Hereby I transfer exclusively to the Journal "Computación y 
Sistemas", published by the Computing Research Center (CIC-IPN),
the Copyright of the aforementioned paper. I also accept that these 
rights will not be transferred to any other publication, in any other 
format, language or other existing means of developing.
I certify that the paper has not been previously disclosed or simultaneo
usly submitted to any other publication, and that it does not contain 
material whose publication would violate the Copyright or other 
proprietary rights of any person, company or institution. I certify that 
I have the permission from the institution or company where I work or 
study to publish this work.
The representative author accepts the responsibility for the publication
of this paper on behalf of each and every one of the authors. 
This transfer is subject to the following conditions:
  • The authors retain all ownership rights (such as patent rights) of this work, except for the publishing rights transferred to the CIC, through this document.
  • Authors retain the right to publish the work in whole or in part in any book they are the authors or publishers. They can also make use of this work in conferences, courses, personal web pages, and so on.
  • Authors may include working as part of his thesis, for non-profit distribution only.