Abstract
This paper presents an implementation of the Variable Neighborhood Search (VNS) metaheuristic for solving the optimization version of the Multidimensional Multi-Way Number Partitioning Problem (MDMWNPP). This problem consists in distributing the vectors of a given sequence into k disjoint subsets such that the sums of each subset form a set of vectors with minimum diameter. The proposed VNS for solving MDMWNPP has a good performance over instances with three and four subsets. A comparative study of results found from this proposed VNS and an implementation of Memetic Algorithm (MA) is carried out, running in the same proportional time interval. Although the average results are different, the statistical tests show that results of the proposed VNS are not significantly better than MA in a set of instances analyzed.
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References
Faria, A.F., de Souza, S.R., Silva, C.A.: Variable neighborhood descent applied to multi-way number partitioning problem. Electron. Notes Discret. Math. 66, 103–110 (2018). https://doi.org/10.1016/j.endm.2018.03.014. 5th International Conference on Variable Neighborhood Search
Gent, I.P., Walsh, T.: Analysis of heuristics for number partitioning. Comput. Intell. 14(3), 430–451 (1998)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. XLV(9), 1563–1581 (1966)
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1969)
Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack problem. J. ACM (JACM) 21(2), 277–292 (1974)
Karmarkar, N., Karp, R.M.: The differencing method of set partition. Report UCB/CSD 81/113, Computer Science Division, University of California, Berkeley, CA (1982)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Proceedings of a Symposium on the Complexity of Computer Computations. The IBM Research Symposia, pp. 85–103. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9
Kojić, J.: Integer linear programming model for multidimensional two-way number partitioning problem. Comput. Math. Appl. 60(8), 2302–2308 (2010)
Korf, R.E.: A complete anytime algorithm for number partitioning. Artif. Intell. 106(2), 181–203 (1998)
Korf, R.E.: Multi-way number partitioning. In: Proceedings of the 21st International Joint Conference on Artifical Intelligence (IJCAI 2009), pp. 538–543 (2009)
Korf, R.E., Schreiber, E.L., Moffitt, M.D.: Optimal sequential multi-way number partitioning. In: Proceedings of the International Symposium on Artificial Intelligence and Mathematics (ISAIM-2014) (2014)
Kratica, J., Kojic, J., Savic, A.: Two metaheuristic approaches for solving multidimensional two-way number partitioning problem. Comput. Oper. Res. 46, 59–68 (2014)
Labs, P.: Geekbench, May 2018. https://browser.geekbench.com/
Mertens, S.: A complete anytime algorithm for balanced number partitioning (1999). http://arxiv.org/abs/cs.DS/9903011
Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)
Moffitt, M.D.: Search strategies for optimal multi-way number partitioning. In: Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence, pp. 623–629. AAAI Press (2013)
Nijenhuis, A., Wilf, H.S.: Combinatorial Algorithms: For Computers and Calculators. Academic Press, Cambridge (2014)
Orlov, M.: Efficient generation of set partitions. Technical report, Faculty of Engineering and Computer Sciences, University of Ulm (2002). http://www.cs.bgu.ac.il/~orlovm/papers/partitions.pdf
Pedroso, J.P., Kubo, M.: Heuristics and exact methods for number partitioning. Eur. J. Oper. Res. 202(1), 73–81 (2010)
Pop, P.C., Matei, O.: A memetic algorithm approach for solving the multidimensional multi-way number partitioning problem. Appl. Math. Modell. 37(22), 9191–9202 (2013)
Rodriguez, F.J., Glover, F., García-Martínez, C., Martí, R., Lozano, M.: GRASP with exterior path-relinking and restricted local search for the multidimensional two-way number partitioning problem. Comput. Oper. Res. 78, 243–254 (2017)
Schreiber, E.L.: Optimal Multi-Way Number Partitioning. Ph.D. thesis, University of California Los Angeles (2014)
Schreiber, E.L., Korf, R.E.: Cached iterative weakening for optimal multi-way number partitioning. In: Proceedings of the Twenty-Eighth Annual Conference on Artificial Intelligence (AAAI-2014), Quebec City, Canada (2014)
Schroeppel, R., Shamir, A.: A \({T}={O}(2^{n/2})\), \({S}={O}(2^{n/4})\) algorithm for certain NP-complete problems. SIAM J. Comput. 10(3), 456–464 (1981)
Sloane, N.: On-Line Encyclopedia of Integer Sequences (1991). https://oeis.org/
Acknowledgements
The authors would like to thank the CAPES Foundation, the Brazilian Council of Technological and Scientific Development (CNPq), the Minas Gerais State Research Foundation (FAPEMIG), the Federal Center of Technological Education of Minas Gerais (CEFET-MG), and the Federal University of Ouro Preto (UFOP) for supporting this research.
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Faria, A.F., de Souza, S.R., Souza, M.J.F., Silva, C.A., Nazário Coelho, V. (2019). A Variable Neighborhood Search Approach for Solving the Multidimensional Multi-Way Number Partitioning Problem. In: Sifaleras, A., Salhi, S., Brimberg, J. (eds) Variable Neighborhood Search. ICVNS 2018. Lecture Notes in Computer Science(), vol 11328. Springer, Cham. https://doi.org/10.1007/978-3-030-15843-9_19
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