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Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines

Authors Martin Koutecký , Johannes Zink

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Author Details

Martin Koutecký
  • Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Johannes Zink
  • Institut für Informatik, Universität Würzburg, Germany

Funding

  • Koutecký, Martin: M. Koutecký was partially supported by Charles University project UNCE/SCI/004, by the project 19-27871X of GA ČR, and by an Israel Science Foundation grant 308/18.

Acknowledgements

We thank the organizers of the HOMONOLO 2019 workshop for providing a warm and stimulating research environment, which gave birth to the initial ideas of this paper. We also thank the anonymous reviewers for their helpful remarks.

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Martin Koutecký and Johannes Zink. Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.18

Abstract

The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (Q|HM|C_max) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model (R|HM|C_max), we show that if either the largest job size p_max, or the number of jobs n are polynomially bounded in the instance size |I|, there are algorithms with complexity |I|^poly(k). Our main result is that this is unlikely to be improved, because Q||C_max is W[1]-hard parameterized by k already when n, p_max, and the numbers describing the speeds are polynomial in |I|; the same holds for R|HM|C_max (without speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives 𝓁₂-norm minimization of the load vector and, partially, sum of weighted completion times ∑ w_j C_j. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (P||C_max) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for Q||C_max this implies that the complexity of P|HM|C_max is the only remaining open case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Scheduling algorithms
Keywords
  • Scheduling
  • cutting stock
  • hardness
  • parameterized complexity

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