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On Minimizing Generalized Makespan on Unrelated Machines

Authors Nikhil Ayyadevara, Nikhil Bansal, Milind Prabhu

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Author Details

Nikhil Ayyadevara
  • University of Michigan, Ann Arbor, MI, USA
Nikhil Bansal
  • University of Michigan, Ann Arbor, MI, USA
Milind Prabhu
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Bansal, Nikhil: Supported in part by the NWO VICI grant 639.023.812.

Cite AsGet BibTex

Nikhil Ayyadevara, Nikhil Bansal, and Milind Prabhu. On Minimizing Generalized Makespan on Unrelated Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.21

Abstract

We consider the Generalized Makespan Problem (GMP) on unrelated machines, where we are given n jobs and m machines and each job j has arbitrary processing time p_{ij} on machine i. Additionally, there is a general symmetric monotone norm ψ_i for each machine i, that determines the load on machine i as a function of the sizes of jobs assigned to it. The goal is to assign the jobs to minimize the maximum machine load. Recently, Deng, Li, and Rabani [Deng et al., 2023] gave a 3 approximation for GMP when the ψ_i are top-k norms, and they ask the question whether an O(1) approximation exists for general norms ψ? We answer this negatively and show that, under natural complexity assumptions, there is some fixed constant δ > 0, such that GMP is Ω(log^δ n) hard to approximate. We also give an Ω(log^{1/2} n) integrality gap for the natural configuration LP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
Keywords
  • Hardness of Approximation
  • Generalized Makespan

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References

  1. Noga Alon, Yossi Azar, Gerhard J Woeginger, and Tal Yadid. Approximation schemes for scheduling on parallel machines. Journal of Scheduling, 1(1):55-66, 1998. Google Scholar
  2. Sanjeev Arora and Carsten Lund. Hardness of approximations. In Approximation algorithms for NP-hard problems, pages 399-446. PWS Publishing Company, Boston, 1997. Google Scholar
  3. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, May 1998. URL: https://doi.org/10.1145/278298.278306.
  4. Yossi Azar and Amir Epstein. Convex programming for scheduling unrelated parallel machines. In Symposium on Theory of Computing, pages 331-337, 2005. URL: https://doi.org/10.1145/1060590.1060639.
  5. Deeparnab Chakrabarty and Chaitanya Swamy. Approximation algorithms for minimum norm and ordered optimization problems. In Symposium on Theory of Computing, pages 126-137, 2019. URL: https://doi.org/10.1145/3313276.3316322.
  6. Deeparnab Chakrabarty and Chaitanya Swamy. Simpler and better algorithms for minimum-norm load balancing. In 27th Annual European Symposium on Algorithms, volume 144 of LIPIcs, pages 27:1-27:12, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.27.
  7. Siu On Chan. Approximation resistance from pairwise-independent subgroups. J. ACM, 63(3):27:1-27:32, 2016. URL: https://doi.org/10.1145/2873054.
  8. Shichuan Deng, Jian Li, and Yuval Rabani. Generalized unrelated machine scheduling problem. In Symposium on Discrete Algorithms (SODA), pages 2898-2916. SIAM, 2023. Google Scholar
  9. Uriel Feige. A threshold of ln n for approximating set cover (preliminary version). In Symposium on the Theory of Computing, pages 314-318, 1996. URL: https://doi.org/10.1145/237814.237977.
  10. Thomas Holenstein. Parallel repetition: simplifications and the no-signaling case. In Symposium on Theory of Computing, pages 411-419, 2007. Google Scholar
  11. Sharat Ibrahimpur and Chaitanya Swamy. Minimum-norm load balancing is (almost) as easy as minimizing makespan. In 48th International Colloquium on Automata, Languages, and Programming, volume 198 of LIPIcs, pages 81:1-81:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.81.
  12. Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in grassmann graph have near-perfect expansion. In Symposium on Foundations of Computer Science, FOCS, pages 592-601, 2018. URL: https://doi.org/10.1109/FOCS.2018.00062.
  13. V. S. Anil Kumar, Madhav V. Marathe, Srinivasan Parthasarathy, and Aravind Srinivasan. A unified approach to scheduling on unrelated parallel machines. J. ACM, 56(5):28:1-28:31, 2009. URL: https://doi.org/10.1145/1552285.1552289.
  14. Jan Karel Lenstra, David B. Shmoys, and Éva Tardos. Approximation algorithms for scheduling unrelated parallel machines. Math. Program., 46:259-271, 1990. URL: https://doi.org/10.1007/BF01585745.
  15. Carsten Lund and Mihalis Yannakakis. On the hardness of approximating minimization problems. Journal of the ACM (JACM), 41(5):960-981, 1994. Google Scholar
  16. Konstantin Makarychev and Maxim Sviridenko. Solving optimization problems with diseconomies of scale via decoupling. J. ACM, 65(6):42:1-42:27, 2018. URL: https://doi.org/10.1145/3266140.
  17. Anup Rao. Parallel repetition in projection games and a concentration bound. In Symposium on Theory of Computing, pages 1-10, 2008. Google Scholar
  18. Ran Raz. A parallel repetition theorem. In Symposium on Theory of computing, pages 447-456, 1995. Google Scholar
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