Abstract
The Maximum Diversity Problem (MDP) requires to extract a subset M of given cardinality from a set N, maximising the sum of the pair-wise diversities between the extracted elements. The MDP has recently been the subject of much research, and several sophisticated heuristics have been proposed to solve it. The present work compares four local search metaheuristics for the MDP, all based on the same Tabu Search procedure, with the aim to identify what additional elements provide the strongest improvement. The four metaheuristics are an Exploring Tabu Search, a Scatter Search, a Variable Neighbourhood Search and a simple Random Restart algorithm. All of them prove competitive with the best algorithms proposed in the literature. Quite surprisingly, the best ones are the simple Random Restart algorithm and a Variable Neighbourhood Search algorithm with an unusual parameter setting, which makes it quite close to random restart. Although this is probably related to the elementary structure of the MDP, it also suggests that, more often than expected, simpler algorithms might be better.
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References
Ahuja RK, Ergun O, Orlin JB and Punnen AP (2002). A survey of very large-scale neighborhood search techniques. Disc Appl Math 123: 75–102.
Andrade PMD, Plastino A, Ochi LS and Martins SL (2003). GRASP for the maximum diversity problem. In Proceedings of the Fifth Metaheuristics International Conference (MIC 2003); Kyoto, Japan.
Aringhieri R and Cordone R (2006). Better and faster solutions for the maximum diversity problem. Note del Polo 93, Università degli Studi di Milano, Crema, April.
Aringhieri R and Dell'Amico M (2005a). Comparing metaheuristic algorithms for SONET network design problems. J Heurist 11: 35–57.
Aringhieri R and Dell'Amico M (2005b). Solution of the SONET ring assignment problem with capacity constraints. In: Rego C and Alidaee B (eds). Metaheuristic Optimization via Memory and Evolution: Tabu Search and Scatter Search. Kluwer Academic Publisher: Dordrecht, pp 93–116.
Aringhieri R, Bruglieri M, Malucelli F and Nonato M (2005). Metaheuristics for a vehicle routing problem on bipartite graphs with distance constraints. In: Proceedings of Sixth Metaheuristics International Conference. August, Wien.
Aringhieri R, Cordone R and Melzani Y (2007). An Ant Colony Optimization approach to the maximum diversity problem. Note del Polo 109, Università degli Studi di Milano, Crema, October.
Aringhieri R, Cordone R and Melzani Y (2008). Tabu search vs. GRASP for the maximum diversity problem. 4OR: Quart J Opns Res 6 (1): 45–60.
Aringhieri R, Bruglieri M and Cordone R (2009). Optimal results and tight bounds for the maximum diversity problem. Found Comput Decis Sci 34 (2): 73–85.
Brimberg J, Mladenović N, Urošević D and Ngai E (2009). Variable neighbourhood search for the heaviest k-subgraph. Comp Opns Res 36: 2885–2891.
Campos V, Duarte A, Gallego M, Gortazar F, Martì R and Piñana E . (2010) Optsicom. http://heur.uv.es/optsicom/.
Cordone R and Wolfler Calvo R (2001). A heuristic for the vehicle routing problem with time windows. J Heurist 7: 107–129.
Dell'Amico M and Trubian M (1998). Solution of large weighted equicut problems. Eur J Opl Res 106: 500–521.
Dorigo M and Stützle T (2004). Ant Colony Optimization. MIT Press: Cambridge, MA.
Duarte A and Martì R (2007). Tabu search and GRASP for the maximum diversity problem. Eur J Opl Res 178: 71–84.
Esposito A, Fiorenzo Catalano MS, Malucelli F and Tarricone L (1998). A new matrix bandwidth reduction algorithm. Opns Res Lett 23: 99–107.
Festa P and Resende MGC (2002). GRASP: An annotated bibliography. In: Ribeiro CC and Hansen P (eds). Essays and Surveys in Metaheuristics. Kluwer Academic Publishers: Dordrecht, pp 325–367.
Gallego M, Duarte A, Laguna M and Martì R (2009). Hybrid heuristics for the maximum diversity problem. Comput Optim Appl 44: 411–426.
Gendreau M, Hertz A and Laporte G (1992). New insertion and postoptimization procedures for the traveling salesman problem. Opns Res 40: 1086–1094.
Gendreau M, Hertz A and Laporte G (1994). A tabu search heuristic for the vehicle routing problem. Mngt Sci 40: 1276–1290.
Ghosh JB (1996). Computational aspects of the maximum diversity problem. Opns Res Lett 19: 175–181.
Glover F and Laguna M (1997). Tabu Search. Kluwer Academic Publishers: Dordrecht.
Glover F, Hersh G and McMillian C (1977). Selecting subset of maximum diversity. MS/IS 77-9, University of Colorado at Boulder.
Glover F, Kuo CC and Dhir KS (1995). A discrete optimization model for preserving biological diversity. Appl Math Modell 19: 696–701.
Glover F, Kuo CC and Dhir KS (1996). Integer programming and heuristic approaches to the Minimum Diversity Problem. J Bus Mngt 4 (1): 93–111.
Hansen P and Mladenović N (2001). Variable neighbourhood search: Principles and applications. Eur J Opns Res 130: 449–467.
Hansen P and Mladenović N (2003). Variable neighborhood search. In: Glover F and Kochenagen G (eds). Handbook of Metaheuristics. Kluwer Academic Publishers: Dordrecht, pp 145–184.
Kleinberg J and Tardos É (2006). Algorithm Design. Addison-Wesley: Reading, MA.
Kochenberger G and Glover F (1999). Title: Diversity data mining. Working Paper Series HCES-03-99, The University of Mississipi.
Kuo CC, Glover F and Dhir KS (1993). Analyzing and modeling the maximum diversity problem by zero-one programming. Decis Sci 24: 1171–1185.
Laguna M and Armentano VA (2005). Lessons from applying and experimenting with scatter search. In: Rego C and Alidaee B (eds). Metaheuristic Optimization via Memory and Evolution: Tabu Search and Scatter Search. Chapter 10, Kluwer Academic Publishers: Dordrecht, pp 229–246.
Laguna M and Martí R (2003). Scatter Search: Methodology and Implementations in C. Kluwer Academic Publishers: Dordrecht.
L'Ecuyer P (1999). Good parameters and implementations for combined multiple recursive random number generators. Opns Res 47: 159–164.
Lin S and Kernighan BW (1973). An effective heuristic algorithm for the traveling-salesman problem. Opns Res 21: 498–516.
Martì R, Gallego M and Duarte A (2010). A branch and bound algorithm for the maximum diversity problem. Eur J Opl Res 200: 36–44.
Palubeckis G (2007). Iterated tabu search for the maximum diversity problem. Appl Math Comput 189: 371–383.
Press WH (1990). Numerical Recipes in C. Cambridge University Press: Cambridge.
Santos LF, Ribeiro MH, Plastino A and Martins SL (2005). A hybrid GRASP with data mining for the maximum diversity problem. In: Blesa MJ, Blum C, Roli A and Sampels M (eds). Hybrid Metaheuristics, Second International Workshop. Volume 3636 of Lecture Notes in Computer Science. Springer: Berlin/Heidelberg, pp 116–127.
Silva GC, Ochi LS and Martins SL (2004). Experimental comparison of greedy randomized adaptive search procedures for the Maximum Diversity Problem. In: CC Ribeiro and SL Martins (eds). Proceedings of the 3rd International Workshop on Efficient and Experimental Algorithms (WEA 2004). Volume 3059 of Lectures Notes on Computer Science, Springer: Berlin/Heidelberg, pp 498–512.
Silva GC, de Andrade MRQ, Ochi LS, Martins SL and Plastino A (2007). New heuristics for the maximum diversity problem. J Heurist 13: 315–336.
Weitz R and Lakshminarayanan S (1998). An empirical comparison of heuristic methods for creating maximally diverse groups. J Opl Res Soc 49: 635–646.
Wilcoxon F (1945). Individual comparisons by ranking methods. Biometrics 1: 80–83.
Acknowledgements
The authors wish to thank Yari Melzani, Gian Paolo Ghilardi, Alberto Ghilardi, Andrea Beretta and Marco Tadini for their help in the computational experience reported in Aringhieri and Cordone (2006) and Aringhieri et al (2007).
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Aringhieri, R., Cordone, R. Comparing local search metaheuristics for the maximum diversity problem. J Oper Res Soc 62, 266–280 (2011). https://doi.org/10.1057/jors.2010.104
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DOI: https://doi.org/10.1057/jors.2010.104