Friendly bin packing instances without Integer Round-up Property
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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.
KeywordsBin packing problem Integer Round-up Property Petersen graph
Mathematics Subject Classification90C11 Mixed integer programming 90C27 Combinatorial optimization 05C15 Coloring of graphs and hypergraphs 05C70 Factorization matching partitioning covering and packing
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