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Scheduling to Approximate Minimization Objectives on Identical Machines

Author Benjamin Moseley

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ACM Subject Classification
  • Theory of computation
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Scheduling
  • LP rounding
  • Approximation Algorithms

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Abstract

This paper considers scheduling on identical machines. The scheduling objective considered in this paper generalizes most scheduling minimization problems. In the problem, there are n jobs and each job j is associated with a monotonically increasing function g_j. The goal is to design a schedule that minimizes sum_{j in [n]} g_{j}(C_j) where C_j is the completion time of job j in the schedule. An O(1)-approximation is known for the single machine case. On multiple machines, this paper shows that if the scheduler is required to be either non-migratory or non-preemptive then any algorithm has an unbounded approximation ratio. Using preemption and migration, this paper gives a O(log log nP)-approximation on multiple machines, the first result on multiple machines. These results imply the first non-trivial positive results for several special cases of the problem considered, such as throughput minimization and tardiness. 
Natural linear programs known for the problem have a poor integrality gap. The results are obtained by strengthening a natural linear program for the problem with a set of covering inequalities we call job cover inequalities. This linear program is rounded to an integral solution by building on quasi-uniform sampling and rounding techniques.

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Benjamin Moseley. Scheduling to Approximate Minimization Objectives on Identical Machines. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 86:1-86:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.86

Author Details

Benjamin Moseley
  • Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA
  • Relational AI, Berkeley, CA, USA

Funding

  • Moseley, Benjamin: Supported in part by a Google Research Award, a Infor Award and NSF Grants CCF-1824303, CCF-1733873 and CCF-1845146.

References

  1. Antonios Antoniadis, Ruben Hoeksma, Julie Meißner, José Verschae, and Andreas Wiese. A QPTAS for the General Scheduling Problem with Identical Release Dates. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 31:1-31:14, 2017. Google Scholar
  2. Nikhil Bansal and Kirk Pruhs. The Geometry of Scheduling. SIAM J. Comput., 43(5):1684-1698, 2014. Google Scholar
  3. Robert D. Carr, Lisa Fleischer, Vitus J. Leung, and Cynthia A. Phillips. Strengthening integrality gaps for capacitated network design and covering problems. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9-11, 2000, San Francisco, CA, USA., pages 106-115, 2000. Google Scholar
  4. Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1576-1585, 2012. Google Scholar
  5. T. C. Edwin Cheng, C. T. Ng, J. J. Yuan, and Zhaohui Liu. Single machine scheduling to minimize total weighted tardiness. European Journal of Operational Research, 165(2):423-443, 2005. Google Scholar
  6. Maurice Cheung, Julián Mestre, David B. Shmoys, and José Verschae. A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems. SIAM J. Discrete Math., 31(2):825-838, 2017. Google Scholar
  7. Maurice Cheung and David B. Shmoys. A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011, Princeton, NJ, USA, August 17-19, 2011. Proceedings, pages 135-146, 2011. Google Scholar
  8. Julia Chuzhoy, Sudipto Guha, Sanjeev Khanna, and Joseph Naor. Machine Minimization for Scheduling Jobs with Interval Constraints. In 45th Symposium on Foundations of Computer Science (FOCS 2004), 17-19 October 2004, Rome, Italy, Proceedings, pages 81-90, 2004. Google Scholar
  9. Dorit S. Hochbaum and David B. Shmoys. Using Dual Approximation Algorithms for Scheduling Problems: Theoretical and Practical Results. In 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21-23 October 1985, pages 79-89, 1985. Google Scholar
  10. Wiebke Höhn and Tobias Jacobs. On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost. ACM Trans. Algorithms, 11(4):25, 2015. Google Scholar
  11. Wiebke Höhn, Julián Mestre, and Andreas Wiese. How Unsplittable-Flow-Covering Helps Scheduling with Job-Dependent Cost Functions. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 625-636, 2014. Google Scholar
  12. Sungjin Im and Benjamin Moseley. Fair Scheduling via Iterative Quasi-Uniform Sampling. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2601-2615, 2017. Google Scholar
  13. E.L. Lawler. A fully polynomial approximation scheme for the total tardiness problem. Operations Research Letters, 1(6):207-208, 1982. URL: http://dx.doi.org/10.1016/0167-6377(82)90022-0.
  14. Retsef Levi, Andrea Lodi, and Maxim Sviridenko. Approximation Algorithms for the Capacitated Multi-Item Lot-Sizing Problem via Flow-Cover Inequalities. Math. Oper. Res., 33(2):461-474, 2008. Google Scholar
  15. David Lo, Liqun Cheng, Rama Govindaraju, Parthasarathy Ranganathan, and Christos Kozyrakis. Heracles: Improving Resource Efficiency at Scale. In Proceedings of the 42Nd Annual International Symposium on Computer Architecture, ISCA '15, pages 450-462, Portland, Oregon, 2015. ACM. URL: http://dx.doi.org/10.1145/2749469.2749475.
  16. Jason Mars, Lingjia Tang, Robert Hundt, Kevin Skadron, and Mary Lou Soffa. Bubble-Up: Increasing Utilization in Modern Warehouse Scale Computers via Sensible Co-locations. In Proceedings of the 44th Annual IEEE/ACM International Symposium on Microarchitecture, MICRO-44, pages 248-259, Porto Alegre, Brazil, 2011. ACM. URL: http://dx.doi.org/10.1145/2155620.2155650.
  17. Paul Marshall, Kate Keahey, and Tim Freeman. Improving Utilization of Infrastructure Clouds. In Proceedings of the 2011 11th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing, CCGRID '11, pages 205-214, Washington, DC, USA, 2011. IEEE Computer Society. URL: http://dx.doi.org/10.1109/CCGrid.2011.56.
  18. Julián Mestre and José Verschae. A 4-approximation for scheduling on a single machine with general cost function. CoRR, abs/1403.0298, 2014. URL: http://arxiv.org/abs/1403.0298.
  19. George L. Nemhauser and Laurence A. Wolsey. Integer and combinatorial optimization. Wiley interscience series in discrete mathematics and optimization. Wiley, 1988. Google Scholar
  20. Tony J. Van Roy and Laurence A. Wolsey. Valid inequalities for mixed 0-1 programs. Discrete Applied Mathematics, 14(2):199-213, 1986. Google Scholar
  21. Wayne E. Smith. Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3:59-66, 1956. Google Scholar
  22. Kasturi R. Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 641-648, 2010. Google Scholar
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