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The Support of Bin Packing Is Exponential

Authors Klaus Jansen , Lis Pirotton , Malte Tutas

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LIPIcs.ESA.2025.48.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Integer programming
  • Theory of computation → Packing and covering problems
Keywords
  • Bin Packing
  • Integer Programming
  • Support

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Abstract

Consider the classical Bin Packing problem with d different item sizes s_i and amounts of items a_i. The support of a Bin Packing solution is the number of differently filled bins. In this work, we show that the lower bound on the support of this problem is 2^Ω(d). Our lower bound matches the upper bound of 2^d given by Eisenbrand and Shmonin [Oper.Research Letters '06] up to a constant factor. This result has direct implications for the time complexity of several Bin Packing algorithms, such as Goemans and Rothvoss [SODA '14], Jansen and Klein [SODA '17] and Jansen and Solis-Oba [IPCO '10]. To achieve our main result, we develop a technique to aggregate equality constrained ILPs with many constraints into an equivalent ILP with one constraint. Our technique contrasts existing aggregation techniques as we manage to integrate upper bounds on variables into the resulting constraint. We believe this technique can be useful for solving general ILPs or the d-dimensional knapsack problem.

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Klaus Jansen, Lis Pirotton, and Malte Tutas. The Support of Bin Packing Is Exponential. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.48

Author Details

Klaus Jansen
  • Department of Computer Science, Kiel University, Germany
Lis Pirotton
  • Department of Computer Science, Kiel University, Germany
Malte Tutas
  • Department of Computer Science, Kiel University, Germany

Funding

This work was supported by the German Research Foundation (DFG) - Project DFG JA 612/27-1.

References

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