Additive Approximation Schemes for Load Balancing Problems

Authors Moritz Buchem, Lars Rohwedder, Tjark Vredeveld, Andreas Wiese



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2021.42.pdf
  • Filesize: 0.61 MB
  • 17 pages

Document Identifiers

Author Details

Moritz Buchem
  • Maastricht University, Maastricht, The Netherlands
Lars Rohwedder
  • EPFL, Lausanne, Switzerland
Tjark Vredeveld
  • Maastricht University, Maastricht, The Netherlands
Andreas Wiese
  • University of Chile, Santiago, Chile

Acknowledgements

We wish to thank José Verschae, Alexandra Lassota and Klaus Jansen for helpful discussions.

Cite AsGet BibTex

Moritz Buchem, Lars Rohwedder, Tjark Vredeveld, and Andreas Wiese. Additive Approximation Schemes for Load Balancing Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.42

Abstract

We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most ε h for some suitable parameter h and any given ε > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p_{max}, the maximum processing time of the jobs. Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ε⋅ p_{max}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
Keywords
  • Load balancing
  • Approximation schemes
  • Parallel machine scheduling

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. N. Alon, Y. Azar, G. J. Woeginger, and T. Yadid. Approximation schemes for scheduling on parallel machines. Journal of Scheduling, 1:55-66, 1998. Google Scholar
  2. N. Alon, A. Shapira, and B. Sudakov. Additive approximation for edge-deletion problems. Annals of Mathematics, 170:37-411, 2009. Google Scholar
  3. C. Annamalai, C. Kalaitzis, and O. Svensson. Combinatorial algorithm for restricted max-min fair allocation. ACM Transactions on Algorithms, 13(3):37:1-37:28, 2017. Google Scholar
  4. A. Asadpour, U. Feige, and A. Saberi. Santa claus meets hypergraph matchings. ACM Transactions on Algorithms, 8:24:1-24:9, 2012. Google Scholar
  5. N. Bansal and M. Sviridenko. The santa claus problem. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, STOC, pages 31-40, 2006. Google Scholar
  6. M. Buchem, L. Rohwedder, T. Vredeveld, and A. Wiese. Additive approximation schemes for load balancing problems. CoRR, abs/2007.09333, 2020. URL: http://arxiv.org/abs/2007.09333.
  7. D. Chakrabarty. Max-min allocation. In M.-Y. Kao, editor, Encyclopedia of Algorithms, pages 1244-1247. Springer US, Boston, MA, 2016. Google Scholar
  8. L. Chen, K. Jansen, and G. Zhang. On the optimality of exact and approximation algorithms for scheduling problems. Journal of Computer and System Sciences, 96:1-32, 2018. Google Scholar
  9. S.-W. Cheng and Y. Mao. Restricted max-min allocation: Approximation and integrality gap. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming, ICALP, pages 38:1-38:13, 2019. Google Scholar
  10. S. Davies, T. Rothvoss, and Y. Zhang. A tale of santa claus, hypergraphs and matroids. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2748-2757, 2020. Google Scholar
  11. F. Eisenbrand and G. Shmonin. Carathéodory bounds for integer cones. Operations Research Letters, 34(5):564-568, 2006. Google Scholar
  12. U. Feige. On allocations that maximize fairness. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 287-293. SIAM, 2008. Google Scholar
  13. M. R. Garey and D. S. Johnson. "strong" NP-completeness results: motivation, examples, and implications. Journal of the ACM, 25:499-508, 1978. Google Scholar
  14. R. L. Graham. Bounds for certain multiprocessing anomalies. Bell System Technical Journal, 45:1563-1581, 1966. Google Scholar
  15. R. L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics, 17:416-429, 1969. Google Scholar
  16. B. Haeupler, B. Saha, and A. Srinivasan. New constructive aspects of the lovász local lemma. Journal of the ACM, 58, 2011. Google Scholar
  17. R. Hoberg and T. Rothvoss. A logarithmic additive integrality gap for bin packing. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2616-2625. SIAM, 2017. Google Scholar
  18. D. S. Hochbaum. Various notions of approximations: Good, better, best and more. Approximation algorithms for NP-hard problems, 1997. Google Scholar
  19. D. S. Hochbaum and D. B. Shmoys. Using dual approximation algorithms for scheduling problems: Theoretical and practical results. Journal of the ACM, 34:144-162, 1987. Google Scholar
  20. K. Jansen. An EPTAS for scheduling jobs on uniform processors: Using an MILP relaxation with a constant number of integral variables. SIAM Journal on Discrete Mathematics, 24:457-485, 2010. Google Scholar
  21. K. Jansen, K.-M. Klein, and J. Verschae. Closing the gap for makespan scheduling via sparsification techniques. Mathematics of Operations Research, 45:1371-1392, 2020. Google Scholar
  22. K. Jansen, S. Kratsch, D. Marx, and I. Schlotter. Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences, 79:39-49, 2013. Google Scholar
  23. K. Jansen and L. Rohwedder. On the configuration-lp of the restricted assignment problem. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2670-2678, 2017. Google Scholar
  24. K. Jansen and L. Rohwedder. A quasi-polynomial approximation for the restricted assignment problem. In Proceedings of the 19th International Conference on Integer Programming and Combinatorial Optimization, IPCO, pages 305-316, 2017. Google Scholar
  25. N. Karmarkar and R. M. Karp. An efficient approximation scheme for the one-dimensional bin-packing problem. In Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, FOCS, pages 312-320. IEEE Computer Society, 1982. Google Scholar
  26. D. Knop and M. Koutecký. Scheduling meets n-fold integer programming. Journal of Scheduling, 21(5):493-503, 2018. Google Scholar
  27. D. Knop, M. Koutecký, and M. Mnich. Combinatorial n-fold integer programming and applications. Mathematical Programming, 184(1):1-34, 2020. Google Scholar
  28. A. Kurpisz, M. Mastrolilli, C. Mathieu, T. Mömke, V. Verdugo, and A. Wiese. Semidefinite and linear programming integrality gaps for scheduling identical machines. Math. Program., 172(1-2):231-248, 2018. Google Scholar
  29. J. K. Lenstra, D. B. Shmoys, and E. Tardos. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, 46:259-271, 1990. Google Scholar
  30. R. J. Lipton, E. Markakis, E. Mossel, and A. Saberi. On approximately fair allocations of indivisible goods. In Proceedings of the 5th ACM Conference on Electronic Commerce, EC, pages 125-131, 2004. Google Scholar
  31. M. Mnich and A. Wiese. Scheduling and fixed-parameter tractability. Mathematical Programming, 154(1-2):533-562, 2015. Google Scholar
  32. T. Ophelders, R. Lambers, F. C. R. Spieksma, and T. Vredeveld. A note on equitable hamiltonian cycles. Discrete Applied Mathematics, page forthcoming, 2021. Google Scholar
  33. T. Rothvoss. Better bin packing approximations via discrepancy theory. SIAM Journal on Computing, 45(3):930-946, 2016. Google Scholar
  34. S. K. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23(1):116-127, 1976. Google Scholar
  35. O. Svensson. Santa claus schedules jobs on unrelated machines. SIAM Journal on Computing, 41(5):1318-1341, 2012. Google Scholar
  36. V. G. Vizing. On an estimate of the chromatic class of a p-graph (in russian). Diskret. Analiz, 3:25-30, 1964. Google Scholar
  37. G. J. Woeginger. A polynomial-time approximation scheme for maximizing the minimum machine completion time. Operations Research Letters, 20:149-154, 1997. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail