Solving maximum independent set by asynchronous distributed hopfield-type neural networks
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 371-388.

We propose a heuristic for solving the maximum independent set problem for a set of processors in a network with arbitrary topology. We assume an asynchronous model of computation and we use modified Hopfield neural networks to find high quality solutions. We analyze the algorithm in terms of the number of rounds necessary to find admissible solutions both in the worst case (theoretical analysis) and in the average case (experimental Analysis). We show that our heuristic is better than the greedy one at 1% significance level.

DOI : https://doi.org/10.1051/ita:2006012
Classification : 68W15,  90C59,  05C69
Mots clés : max independent set, hopfield networks, asynchronous distributed algorithms
@article{ITA_2006__40_2_371_0,
author = {Grossi, Giuliano and Marchi, Massimo and Posenato, Roberto},
title = {Solving maximum independent set by asynchronous distributed hopfield-type neural networks},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {371--388},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
doi = {10.1051/ita:2006012},
zbl = {1112.68119},
mrnumber = {2252645},
language = {en},
url = {http://www.numdam.org/item/ITA_2006__40_2_371_0/}
}
Grossi, Giuliano; Marchi, Massimo; Posenato, Roberto. Solving maximum independent set by asynchronous distributed hopfield-type neural networks. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 371-388. doi : 10.1051/ita:2006012. http://www.numdam.org/item/ITA_2006__40_2_371_0/

[1] T. Bäck and S. Khuri, An evolutionary heuristic for the maximum independent set problem, in Proc. First IEEE Conf. Evolutionary Computation, IEEE World Congress on Computational Intelligence, edited by Z. Michalewicz, J.D. Schaffer, H.-P. Schwefel, D.B. Fogel and H. Kitano, Orlando FL, June 27-29. IEEE Press, Piscataway NJ 2 (1994) 531-535.

[2] S. Basagni, Finding a maximal weighted independent set in wireless networks. Telecommunication Systems, Special Issue on Mobile Computing and Wireless Networks 18 (2001) 155-168. | Zbl 1020.68005

[3] R. Battiti and M. Protasi, Reactive local search for the maximum clique problem. Algorithmica 29 (2001) 610-637. | Zbl 0985.68016

[4] A. Bertoni, P. Campadelli and G. Grossi, A neural algorithm for the maximum clique problem: Analysis, experiments and circuit implementation. Algoritmica 33 (2002) 71-88. | Zbl 0994.68002

[5] I.M. Bomze, M. Budinich, M. Pelillo and C. Rossi, Annealed replication: A new heuristic for the maximum clique problem. Discrete Appl. Math. 121 (2000) 27-49. | Zbl 1019.90032

[6] M. Dorigo, G. Di Caro and L.M. Gambardella, Ant algorithms for discrete optimization. Artificial Life 5 (1996) 137-172.

[7] U. Feige, S. Goldwasser, S. Safra, L. Lovàsz and M. Szegedy, Approximating clique is almost NP-complete, in Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science (1991) 2-12.

[8] T.A. Feo, M.G.C. Resende and S.H. Smith, Greedy randomized adaptive search procedure for maximum independent set. Oper. Res. 41 (1993). | Zbl 0815.90121

[9] M. Gerla and J.T.-C. Tsai, Multicluster, mobile, multimedia radio network. Wireless Networks 1 (1995) 255-265.

[10] W. Goddard, S.T. Hedetniemi, D.P. Jacobs and P.K. Srimani, Self-stabilizing protocols for maximal matching and maximal independent sets for ad hoc networks, in Workshop on Advances in Parallel and Distributed Computational Models (2003).

[11] G. Grossi and R. Posenato, A distributed algorithm for max independent set problem based on Hopfield networks, in Neural Nets: 13th Italian Workshop on Neural Nets (WIRN 2002), edited by M. Marinaro and R. Tagliaferri. Springer-Verlag. Lect. Notes Comput. Sci. 2486 (2002) 64-74. | Zbl 1028.68592

[12] M.M. Halldórsson and J. Radhakrishnan, Greedy is good: Approximating independent sets in sparse and bounded-degree graphs. Algorithmica 18 (1997) 145-163. | Zbl 0866.68077

[13] J. Håstad, Clique is hard to approximate within ${n}^{1-ϵ}$, in Proc. of the 37rd Annual IEEE Symposium on the Foundations of Computer Science (1996) 627-636.

[14] J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, in Proceedings of the National Academy of Sciences of the United States of America 79 (1982) 2554-2558.

[15] L. Ji-Cherng and T.C. Huang, An efficient fault-containing self-stabilizing algorithm for finding a maximal independent set. IEEE Transactions on Parallel and Distributed Systems 14 (2003) 742-754.

[16] D.S. Johnson and M.A. Trick, Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society (1996). | MR 1423138 | Zbl 0851.00080

[17] R.M. Karp, Reducibility among Combinatorial Problems. Complexity of Computer Computations. Plenum Press, New York (1972) 85-103.

[18] G. Leguizam'On, Z. Michalewicz and M. Schütz, An ant system for the maximum independent set problem, in Proceedings of VII Argentine Congress of Computer Science (CACIC 2001) (2001).

[19] S.Z. Li, Improving convergence and solution quality of Hopfield-type neural networks with augmented Lagrange multipliers. IEEE Trans. Neural Networks 7 (1996) 1507-1516.

[20] E. Marchiori, A simple heuristic based genetic algorithm for the maximum clique problem, in Proc. ACM Symp. Appl. Comput (1998) 366-373.

[21] C.H. Papadimitriou, Computational Complexity. Addison-Wesley (1994). | MR 1251285 | Zbl 0833.68049