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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 Iori
  • Romeo Rizzi
Full Length Paper Series B

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.

Keywords

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 

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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
  • 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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