Abstract
We introduce and asses several Divide-and-Conquer heuristic strategies, aimed at solving large instances of the 0–1 Minimization Knapsack Problem. The method subdivides a large problem in two smaller ones (or recursive iterations of the same procedure), in order to lower down the global computational complexity of the original problem, at the expense of a moderate loss of quality in the solution. Theoretical mathematical results are presented to assure a successful algorithmic application of the method and to suggest the potential strategies for its implementation. In contrast, due to the lack of theoretical results, the solution’s quality deterioration is measured empirically by means of Monte Carlo simulations for several types and values of the chosen strategies. Finally, introducing parameters of efficiency we suggest the best strategies depending on the data input.
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Acknowledgements
The first Author wishes to thank Universidad Nacional de Colombia, Sede Medellín (https://medellin.unal.edu.co) for supporting the production of this work through the project Hermes 45713, as well as granting access to Gauss Server, financed by “Proyecto Plan 150x150 Fomento de la cultura de evaluación continua a través del apoyo a planes de mejoramiento de los programas curriculares”, where the numerical experiments were executed. The second Author wishes to thank Universidad EAFIT (http://www.eafit.edu.co) for its financial support as MSc student, through the Internal Grant 819156 “Modelos matemáticos y métodos de solución a un tipo de problema logístico que involucha agrupamiento de clientes, distribución y ruteo”. The authors also wish to thank the anonymous referees whose meticulous review and insightful suggestions enhanced substantially the quality of this work. Special thanks to Professor Daniel Cabarcas (https://sites.google.com/a/unal.edu.co/dcabarc/) from Universidad Nacional de Colombia, Sede Medellín for his help in understanding and running the code COMBO from Martello et al. (1999).
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Morales, F.A., Martínez, J.A. Analysis of Divide-and-Conquer strategies for the 0–1 minimization knapsack problem. J Comb Optim 40, 234–278 (2020). https://doi.org/10.1007/s10878-020-00584-2
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DOI: https://doi.org/10.1007/s10878-020-00584-2