Mathematical Programming

, Volume 150, Issue 1, pp 5–17 | Cite as

Friendly bin packing instances without Integer Round-up Property

  • Alberto Caprara
  • Mauro Dell’Amico
  • José Carlos Díaz-Díaz
  • Manuel IoriEmail author
  • Romeo Rizzi
Full Length Paper Series B


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.


Bin packing problem Integer Round-up Property Petersen graph 

Mathematics Subject Classification

90C11 Mixed integer programming  90C27 Combinatorial optimization 05C15 Coloring of graphs and hypergraphs  05C70 Factorization matching partitioning covering and packing 


  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Valério deCarvalho, J.: LP models for bin packing and cutting stock problems. Eur. J. Oper. Res. 141, 253–273 (2002)CrossRefGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting stock problem. Oper. Res. 9, 849–859 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gilmore, P., Gomory, R.: A linear programming approach to the cutting stock problem: part II. Oper. Res. 11, 863–888 (1963)CrossRefzbMATHGoogle Scholar
  6. 6.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Caprara, A., Monaci, M.: Bidimensional packing by bilinear programming. Math. Program. 118, 75–108 (2009)Google Scholar
  9. 9.
    Marcotte, O.: An instance of the cutting stock problem for which the rounding property does not hold. Oper. Res. Lett. 4, 239–243 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chan, L., Simchi-Levi, D., Bramel, J.: Worst-case analyses, linear programming and the bin-packing problem. Math. Program. 83, 213–227 (1998)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Scheitauer, G.: On the maxgap problem for cutting stock. J. Inform. Process. Cybernet. 30, 111–117 (1994)Google Scholar
  12. 12.
    Eisenbrand, F., Pálvölgyi, D., Rothvoß, T.: Bin packing via discrepancy of permutations. ACM Trans. Algorithms 9, 24:1–24:15 (2013)CrossRefGoogle Scholar
  13. 13.
    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)MathSciNetGoogle Scholar
  14. 14.
    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)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • Alberto Caprara
    • 1
  • Mauro Dell’Amico
    • 2
  • José Carlos Díaz-Díaz
    • 2
  • Manuel Iori
    • 2
    Email author
  • Romeo Rizzi
    • 3
  1. 1.BolognaItaly
  2. 2.Department of Sciences and Methods for EngineeringUniversity of Modena and Reggio EmiliaReggio EmiliaItaly
  3. 3.Department of Computer ScienceUniversity of VeronaVeronaItaly

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