Abstract
In this article, the concept of conditioning in integer programming is extended to the concept of a complexity index. A complexity index is a measure through which the execution time of an exact algorithm can be predicted. We consider the multidimensional knapsack problem with instances taken from the OR-library and MIPLIB. The complexity indices we developed correspond to the eigenvalues of a Dikin matrix placed in the center of a polyhedron defined by the constraints of the problem relaxed from its binary variable formulation. The lower and higher eigenvalues, as well as their ratio, which we have defined as the slenderness, are used as complexity indices. The experiments performed show a good linear correlation between these indices and a low execution time of the Branch and Bound algorithm using the standard version of CPLEX® 12.2. The correlation coefficient obtained ranges between 39 and 60% for the various data regressions, which proves a medium to strong correlation.
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References
Angelelli E, Mansini R, Speranza MG, Grazia (2010) Kernel search: a general heuristic for the multi-dimensional knapsack problem. Comput Oper Res 37(11):2017–2026. https://doi.org/10.1016/j.cor.2010.02.002
Beasley JE (1990) OR-Library: distributing test problems by electronic mail. J Oper Res Soc 41(11):1069–1072. https://doi.org/10.1057/jors.1990.166
Bolat A (2003) A mathematical model for selecting mixed models with due dates. Int J Prod Res 41(5):897–918. https://doi.org/10.1080/00207540210163892
Boussier S, Vasquez M, Vimont Y, Hanafi S, Michelon P (2010) A multilevel search strategy for the 0–1 multidimensional knapsack problem. Discrete Appl Math 158(2):97–109
Cherbaka N, Meller R (2008) Single-plant sourcing decisions with a multidimensional knapsack model. J Manuf Syst 27:7–18. https://doi.org/10.1016/j.jmsy.2008.07.001
Chu PC, Beasley JE (1998) A genetic algorithm for the multidimensional knapsack problem. J Heuristics 4(1):63–86. https://doi.org/10.1023/A:1009642405419
Cvetkovic D (2012) Complexity indices for the travelling salesman problem and data mining. Trans Comb 1(1):35–43
Freund R, Vera J (2003) On the complexity of estimating condition measures for conic convex optimization. Math Oper Res 28:625–664
Grandón J, Derpich I (2011) A heuristic for the multi-knapsack problem. WSEAS Trans Math 10(3):95–104
James RJ, Nakagawa Y (2005) Enumeration methods for repeatedly solving multidimensional knapsack sub-problems. IEICE Trans Inf Syst 10:2329–2340
Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, Berlin
Ko K, Orponen P, Schoning U, Watanabe O (1986) What is a hard instance of a computational problem? In: Selman AL (ed) Structure in complexity theory. Lecture notes in computer science. Springer, Berlin, pp 197–217. https://doi.org/10.1007/3-540-16486-3_99
Lalami M, Elkihel M, Baz D, Boyer V (2012) A procedure-based heuristic for 0–1 multiple knapsack problems. Int J Math Oper Res 4(3):214–224
Magazine M, Oguz O (1984) A heuristic algorithm for the multidimensional zero-one knapsack problem. Eur J Oper Res 16(3):319–326
Mansini R, Speranza MG (2002) A multidimensional knapsack model for asset-backed securitization. J Oper Res Soc 53(8):822–832. https://doi.org/10.1057/palgrave.jors.260140
Mansini R, Speranza M (2012) CORAL: an exact algorithm for the multidimensional knapsack problem. INFORMS J Comput 24(3):399–415. https://doi.org/10.1287/ijoc.1110.0460
Martello S, Pisinger D, Toth P (2000) New trends in exact algorithms for the 0–1 knapsack problem. Eur J Oper Res 123(1):325–332
Reyck BD, Herroelen W (1996) On the use of the complexity index as a measure of complexity in activity networks. Eur J Oper Res 91:347–366
Senju S, Toyoda Y (1968) An approach to linear programming with 0–1 variables. Manag Sci 15:196–207
Song Y, Zhang C, Fang Y (2008). Multiple multidimensional knapsack problem and its applications in cognitive radio networks. In: Military communications conference IEEE
Vera J, Derpich I (2006) Incorporating condition measures in the context of combinatorial optimization. SIAM J Optim 16(4):965–985
Volgenant A, Zwiers I (2007) Partial enumeration in heuristics for some combinatorial optimization problems. J Oper Res Soc 58(1):73–79. https://doi.org/10.1057/palgrave.jors.2602102
Wilbaut C, Salhi S, Hanafi S (2009) An iterative variable-based fixation heuristic for the 0–1 multidimensional knapsack problem. Eur J Oper Res 199:339–348
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The authors are very grateful to the DICYT (Scientific and Technological Research Office), Project No. 061317DC, and the Industrial Engineering (IE) Department of the University of Santiago of Chile for their support in this work.
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Derpich, I., Herrera, C., Sepúlveda, F. et al. Complexity indices for the multidimensional knapsack problem. Cent Eur J Oper Res 29, 589–609 (2021). https://doi.org/10.1007/s10100-018-0569-0
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DOI: https://doi.org/10.1007/s10100-018-0569-0