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ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines

Author Lars Rohwedder

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LIPIcs.ICALP.2025.126.pdf
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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Mathematical optimization
Keywords
  • Scheduling
  • Integer Programming

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Abstract

Given n jobs with processing times p₁,...,p_n ∈ ℕ and m ≤ n machines with speeds s₁,...,s_m ∈ ℕ our goal is to allocate the jobs to machines minimizing the makespan. We present an algorithm that solves the problem in time p_{max}^{O(d)} ⋅ n, where p_{max} is the maximum processing time and d ≤ p_{max} is the number of distinct processing times. This is essentially the best possible due to a lower bound based on the exponential time hypothesis (ETH).
Our result improves over prior works that had a quadratic term in d in the exponent and answers an open question by Koutecký and Zink. The algorithm is based on integer programming techniques combined with novel ideas from modular arithmetic. It can also be implemented efficiently for the more compact high-multiplicity instance encoding.

Cite As Get BibTex

Lars Rohwedder. ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 126:1-126:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.126

Author Details

Lars Rohwedder
  • University of Southern Denmark (SDU), Odense, Denmark

Funding

Supported by Dutch Research Council (NWO) project “The Twilight Zone of Efficiency: Optimality of Quasi-Polynomial Time Algorithms” [grant number OCEN.W.21.268].

References

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