Abstract
We propose a multivariable continuous polynomial optimization formulation to find arbitrary maximal independent sets of any size for any graph. A local optima of the optimization problem yields a maximal independent set, while the global optima yields a maximum independent set. The solution is two phases. The first phase is listing all the maximal cliques of the graph and the second phase is solving the optimization problem. We believe that our algorithm is efficient for sparse graphs, for which there exist fast algorithms to list their maximal cliques. Our algorithm was tested on some of the DIMACS maximum clique benchmarks and produced results efficiently. In some cases our algorithm outperforms other algorithms, such as cliquer.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Östergård, P.R.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120(1), 197–207 (2002)
Fang, Z., Li, C.-M., Xu, K.: An exact algorithm based on maxsat reasoning for the maximum weight clique problem. J. Artif. Intell. Res. 55, 799–833 (2016)
Tavares, W.A., Neto, M.B.C., Rodrigues, C.D., Michelon, P.: Um algoritmo de branch and bound para o problema da clique máxima ponderada. In: Proceedings of XLVII SBPO, vol. 1 (2015)
Shimizu, S., Yamaguchi, K., Saitoh, T., Masuda, S.: Fast maximum weight clique extraction algorithm: optimal tables for branch-and-bound. Discrete Appl. Math. 223, 120–134 (2017)
Shor, N.Z.: Dual quadratic estimates in polynomial and Boolean programming. Ann. Oper. Res. 25(1), 163–168 (1990)
Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17(4), 533–540 (1965)
Harant, J.: Some news about the independence number of a graph. Discussiones Math. Graph Theory 20(1), 71–79 (2000)
Harant, J., Pruchnewski, A., Voigt, M.: On dominating sets and independent sets of graphs. Comb. Probab. Comput. 11, 1–10 (1993)
Pardalos, P., Gibbons, L., Hearn, D.: A continuous based heuristic for the maximum clique problem. Technical Report, University of Michigan, Ann Arbor, MI, United States (1994)
Jain, K., Padhye, J., Padmanabhan, V.N., Qiu, L.: Impact of interference on multi-hop wireless network performance. Wireless Netw. 11(4), 471–487 (2005)
Eppstein, D., Löffler, M., Strash, D.: Listing all maximal cliques in sparse graphs in near-optimal time. In: International Symposium on Algorithms and Computation, pp. 403–414. Springer (2010)
Acknowledgment
Maher Heal thanks Dr. Una Benlic for advice on the maximal independent set and maximal clique problems while working on this research.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Heal, M., Li, J. (2020). Finding the Maximal Independent Sets of a Graph Including the Maximum Using a Multivariable Continuous Polynomial Objective Optimization Formulation. In: Arai, K., Kapoor, S., Bhatia, R. (eds) Intelligent Computing. SAI 2020. Advances in Intelligent Systems and Computing, vol 1228. Springer, Cham. https://doi.org/10.1007/978-3-030-52249-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-52249-0_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-52248-3
Online ISBN: 978-3-030-52249-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)