Abstract
It is well known that the gap between the optimal values of bin packing and fractional bin packing, if the latter is rounded up to the closest integer, is almost always null. Known counterexamples to this for integer input values involve fairly large numbers. Specifically, the first one was derived in 1986 and involved a bin capacity of the order of a billion. Later in 1998 a counterexample with a bin capacity of the order of a million was found. In this paper we show a large number of counterexamples with bin capacity of the order of a hundred, showing that the gap may be positive even for numbers which arise in customary applications. The associated instances are constructed starting from the Petersen graph and using the fact that it is fractionally, but not integrally, 3-edge colorable.
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References
Coffman, E., Csirik, J., Galambos, G., Martello, S., Vigo, D.: Bin packing approximation algorithms: survey and classification. In: Du, D.Z., Pardalos, P., Graham, R. (eds.) Handbook of Combinatorial Optimization, 2nd edn, pp. 455–531. Springer, Berlin (2013)
Valério deCarvalho, J.: LP models for bin packing and cutting stock problems. Eur. J. Oper. Res. 141, 253–273 (2002)
Clautiaux, F., Alves, C., Valério de Carvalho, J.: A survey of dual-feasible functions for bin-packing problems. Ann. Oper. Res. 179(1), 317–342 (2009)
Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting stock problem. Oper. Res. 9, 849–859 (1961)
Gilmore, P., Gomory, R.: A linear programming approach to the cutting stock problem: part II. Oper. Res. 11, 863–888 (1963)
Rietz, J., Scheithauer, G., Terno, J.: Families of non-IRUP instances of the one-dimensional cutting stock problem. Discrete Appl. Math. 121, 229–245 (2002)
Rietz, J., Scheithauer, G., Terno, J.: Tighter bounds for the gap and non-IRUP constructions in the one-dimensional cutting stock problem. Optim. J. Math. Program. Oper. Res. 51, 927–963 (2002)
Caprara, A., Monaci, M.: Bidimensional packing by bilinear programming. Math. Program. 118, 75–108 (2009)
Marcotte, O.: An instance of the cutting stock problem for which the rounding property does not hold. Oper. Res. Lett. 4, 239–243 (1986)
Chan, L., Simchi-Levi, D., Bramel, J.: Worst-case analyses, linear programming and the bin-packing problem. Math. Program. 83, 213–227 (1998)
Scheitauer, G.: On the maxgap problem for cutting stock. J. Inform. Process. Cybernet. 30, 111–117 (1994)
Eisenbrand, F., Pálvölgyi, D., Rothvoß, T.: Bin packing via discrepancy of permutations. ACM Trans. Algorithms 9, 24:1–24:15 (2013)
Dell’Amico, M., Iori, M., Monaci, M., Martello, S.: Heuristic and exact algorithms for the identical parallel machine scheduling problem. INFORMS J. Comput. 30, 333–344 (2006)
Belov, G., Scheithauer, G.: A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting. Eur. J. Oper. Res. 171, 85–106 (2006)
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Alberto Caprara died in April 2012 while climbing one of his beloved mountains. His contribution to the core part of this paper has been fundamental for the success of our research.
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Caprara, A., Dell’Amico, M., Díaz-Díaz, J.C. et al. Friendly bin packing instances without Integer Round-up Property. Math. Program. 150, 5–17 (2015). https://doi.org/10.1007/s10107-014-0791-z
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DOI: https://doi.org/10.1007/s10107-014-0791-z