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Improved filtering for the bin-packing with cardinality constraint

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Abstract

Previous research shows that a cardinality reasoning can improve the pruning of the bin-packing constraint. We first introduce a new algorithm, called BPCFlow, that filters both load and cardinality bounds on the bins, using a flow reasoning similar to the Global Cardinality Constraint. Moreover, we detect impossible assignments of items by combining the load and cardinality of the bins, using a method to detect items that are either ”too-big” or ”too-small”. This method is adapted to two previously existing filtering techniques along with BPCFlow, creating three new propagators. We then experiment the four new algorithms on Balanced Academic Curriculum Problem and Tank Allocation Problem instances. BPCFlow is shown to be stronger than previously existing filtering, and more computationally intensive. We show that the new filtering is useful on a small number of hard instances, while being too expensive for general use. Our results show that the introduced ”too-big/too-small” filtering can most of the time drastically reduce the size of the search tree and the computation time. This method is profitable in 88% of the tested instances.

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Notes

  1. For simplicity, we use in the remaining of this paper items weights as a way to identify items

  2. which itself subsumes SimpleBPC

  3. 3i. e. item i 0will be assigned to bin j after cancelation of the cycle

  4. 4Note that u and v can be confounded with i 0, k 0, i 1or k 1,which does not change the proof.

  5. the worst case being a complete graph

  6. A small variation of this algorithm, where the dichotomic search is replaced by a linear one, leading to a complexity of \(\mathcal {O}(m^{3}+m^{2}n^{2})\), was also implemented. Its performances are very close to its dichotomic counterpart.

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Acknowledgements

We thank the anonymous reviewer for suggesting the idea of using a dichotomic search in Algorithm 3.

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Authors and Affiliations

  1. UCLouvain, Louvain-la-Neuve, Belgium

    Guillaume Derval & Pierre Schaus

  2. University of Nice Sophia-Antipolis, Nice, France

    Jean-Charles Régin

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  1. Guillaume Derval
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  2. Jean-Charles Régin
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  3. Pierre Schaus
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Correspondence to Guillaume Derval.

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Derval, G., Régin, JC. & Schaus, P. Improved filtering for the bin-packing with cardinality constraint. Constraints 23, 251–271 (2018). https://doi.org/10.1007/s10601-017-9278-x

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