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Faster Exponential Algorithms for Multi-Machine Scheduling Problems

Authors Anubhav Dhar , Anita Dürr , Ahmed Ghazy , Jakob Greilhuber , Karol Węgrzycki

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Scheduling
  • exact algorithms
  • exponential-time algorithms

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Abstract

Minimizing the weighted completion times (P ‖ Σ w_j C_j) and weighted number of tardy jobs (P ‖ Σ w_j U_j) on multiple identical machines are two classical NP-hard scheduling problems. As shown by Lenté et al. (2014), both problems can be solved in time 𝒪^⋆(3ⁿ). In this paper, we improve these bounds to 𝒪(2.755ⁿ) and 𝒪^⋆(2ⁿ), respectively. Our algorithm for P ‖ Σ w_j C_j exploits the meet-in-the-middle paradigm and an efficient data structure answering linear programming queries. Additionally, when the number of machines is at most 6, we show that the running time for P ‖ Σ w_j C_j can further be improved.
Both scheduling problems are generalizations of the classical Bin Packing problem, which can be solved in 𝒪^⋆(2ⁿ) time. Improving this running time is an important open question. We show that, when assuming the Asymptotic Rank Conjecture (ARC), Bin Packing can be solved in time 𝒪((2-ε)ⁿ) for some ε > 0. Our algorithm makes use of two main ingredients: the recent 𝒪((2-ε)ⁿ)-time algorithm of Nederlof et al. [SICOMP'23] for Bin Packing when the number of bins is a fixed constant, and the 𝒪((2-ε)ⁿ)-time algorithm of Björklund et al. [SODA'25] for special instances of the 3-way Partitioning problem when assuming ARC.

Cite As Get BibTex

Anubhav Dhar, Anita Dürr, Ahmed Ghazy, Jakob Greilhuber, and Karol Węgrzycki. Faster Exponential Algorithms for Multi-Machine Scheduling Problems. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 53:1-53:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.53

Author Details

Anubhav Dhar
  • Max Planck Institute for Informatics, Saarbrücken, Germany
  • Saarland University, Saarbrücken, Germany
Anita Dürr
  • ETH, Zurich, Switzerland
Ahmed Ghazy
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
  • Saarland University, Saarbrücken, Germany
Jakob Greilhuber
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
  • Saarland University, Saarbrücken, Germany
Karol Węgrzycki
  • Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

  • Dhar, Anubhav: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) grant number 559177164.
  • Dürr, Anita: Part of this work was done while affiliated to Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany, where this work was part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 850979).
  • Węgrzycki, Karol: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) grant number 559177164.

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