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A Tight Double-Exponential Lower Bound for High-Multiplicity Bin Packing

Authors Klaus Jansen , Felix Ohnesorge , Lis Pirotton

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Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Bin Packing
  • Lower Bound
  • Computational Complexity
  • ETH

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Abstract

Consider a high-multiplicity Bin Packing instance I with d distinct item types. In 2014, Goemans and Rothvoss gave an algorithm with runtime (|I|²)^O(d) for this problem [SODA'14], where |I| denotes the encoding length of the instance I. Although Jansen and Klein [SODA'17] later developed an algorithm that improves upon this runtime in a special case, it has remained a major open problem by Goemans and Rothvoss [J.ACM'20] whether the doubly exponential dependency on d is necessary.
We solve this open problem by showing that unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm solving the high-multiplicity Bin Packing problem in time (|I|²)^o(d). To prove this, we introduce a novel reduction from 3-SAT. The core of our construction is efficiently encoding all information from a 3-SAT instance with n variables into an ILP with O(log n) variables and constraints.
This result confirms that the Goemans and Rothvoss algorithm is essentially best-possible for Bin Packing parameterized by the number d of item sizes in the context of XP time algorithms.

Cite As Get BibTex

Klaus Jansen, Felix Ohnesorge, and Lis Pirotton. A Tight Double-Exponential Lower Bound for High-Multiplicity Bin Packing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 116:1-116:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.116

Author Details

Klaus Jansen
  • Kiel University, Germany
Felix Ohnesorge
  • Kiel University, Germany
Lis Pirotton
  • Kiel University, Germany

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project number 453769249.

Acknowledgements

We thank Alberto Del Pia and Timo Berthold for discussions on constraint linearization and Stefan Weltge for pointing us to the references [Volker Kaibel and Stefan Weltge, 2015; Alexander Schrijver, 1999]. The idea for this work originated during a visit of Klaus Jansen to the MPI - INF Saarbrücken in 2024.

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