Minimum-Norm Load Balancing Is (Almost) as Easy as Minimizing Makespan

Authors Sharat Ibrahimpur , Chaitanya Swamy

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Sharat Ibrahimpur
  • Department of Combinatorics and Optimization, University of Waterloo, Canada
Chaitanya Swamy
  • Department of Combinatorics and Optimization, University of Waterloo, Canada

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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 (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 81:1-81:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We consider the minimum-norm load-balancing (MinNormLB) problem, wherein there are n jobs, each of which needs to be assigned to one of m machines, and we are given the processing times {p_{ij}} of the jobs on the machines. We also have a monotone, symmetric norm f:ℝ^m → ℝ_{≥ 0}. We seek an assignment σ of jobs to machines that minimizes the f-norm of the induced load vector load->_σ ∈ ℝ_{≥ 0}^m, where load_σ(i) = ∑_{j:σ(j) = i}p_{ij}. This problem was introduced by [Deeparnab Chakrabarty and Chaitanya Swamy, 2019], and the current-best result for MinNormLB is a (4+ε)-approximation [Deeparnab Chakrabarty and Chaitanya Swamy, 2019]. In the stochastic version of MinNormLB, the job processing times are given by nonnegative random variables X_{ij}, and jobs are independent; the goal is to find an assignment σ that minimizes the expected f-norm of the induced random load vector. We obtain results that (essentially) match the best-known guarantees for deterministic makespan minimization (MinNormLB with 𝓁_∞ norm). For MinNormLB, we obtain a (2+ε)-approximation for unrelated machines, and a PTAS for identical machines. For stochastic MinNormLB, we consider the setting where the X_{ij}s are Poisson random variables, denoted PoisNormLB. Our main result here is a novel and powerful reduction showing that, for any machine environment (e.g., unrelated/identical machines), any α-approximation algorithm for MinNormLB in that machine environment yields a randomized α(1+ε)-approximation for PoisNormLB in that machine environment. Combining this with our results for MinNormLB, we immediately obtain a (2+ε)-approximation for PoisNormLB on unrelated machines, and a PTAS for PoisNormLB on identical machines. The latter result substantially generalizes a PTAS for makespan minimization with Poisson jobs obtained recently by [Anindya De et al., 2020].

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Approximation algorithms
  • Load balancing
  • Minimum-norm optimization
  • LP rounding


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  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. Yossi Azar and Amir Epstein. Convex programming for scheduling unrelated parallel machines. In Proceedings of the 37th STOC, pages 331-337, 2005. Google Scholar
  3. Deeparnab Chakrabarty. Personal Communication, 2021. Google Scholar
  4. Deeparnab Chakrabarty and Chaitanya Swamy. Approximation algorithms for minimum norm and ordered optimization problems. In Proceedings of the 51st STOC, pages 126-137, 2019. Google Scholar
  5. Deeparnab Chakrabarty and Chaitanya Swamy. Simpler and better algorithms for minimum-norm load balancing. In Proceedings of the 27th ESA, pages 27:1-27:12, 2019. Google Scholar
  6. Anindya De, Sanjeev Khanna, Huan Li, and Hesam Nikpey. An efficient PTAS for stochastic load balancing with poisson jobs. In Proceedings of the 47th ICALP, pages 37:1-37:18, 2020. Google Scholar
  7. Ashish Goel and Piotr Indyk. Stochastic load balancing and related problems. In Proceedings of the 40th FOCS, pages 579-586, 1999. Google Scholar
  8. Anupam Gupta, Amit Kumar, Viswanath Nagarajan, and Xiangkun Shen. Stochastic load balancing on unrelated machines. In Proceedings of the 29th SODA, pages 1274-1285, 2018. Google Scholar
  9. G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge Univ Press, 1934. Google Scholar
  10. D. S. Hochbaum and D. B. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach. SICOMP, 17(3):539-551, 1988. Google Scholar
  11. Dorit Hochbaum and David B. Shmoys. Using dual approximation algorithms for scheduling problems: practical and theoretical results. J. ACM, 34(1):144-162, 1987. Google Scholar
  12. Sharat Ibrahimpur and Chaitanya Swamy. Approximation algorithms for stochastic minimum-norm combinatorial optimization. In Proceedings of the 61st FOCS, pages 966-977, 2020. Google Scholar
  13. K. Jansen. An EPTAS for scheduling jobs on uniform processors: Using an MILP relaxation with a constant number of integral variables. In Proc. 36th ICALP, pages 562-573, 2009. Google Scholar
  14. Klaus Jansen, Kim-Manuel Klein, and José Verschae. Closing the Gap for Makespan Scheduling via Sparsification Techniques. In Proceedings of the 43rd ICALP, pages 72:1-72:13, 2016. Google Scholar
  15. Jon M. Kleinberg, Yuval Rabani, and Éva Tardos. Allocating bandwidth for bursty connections. SIAM J. Comput., 30(1):191-217, 2000. Google Scholar
  16. V. S. Kumar, Madhav V Marathe, S. Parthasarathy, and A. Srinivasan. A unified approach to scheduling on unrelated parallel machines. Journal of the ACM, 56(5):28, 2009. Google Scholar
  17. Jan Karel Lenstra, David B. Shmoys, and Éva Tardos. Approximation algorithms for scheduling unrelated parallel machines. Math. Program., 46:259-271, 1990. Google Scholar
  18. Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538-548, 1983. Google Scholar
  19. André Linhares, Neil Olver, Chaitanya Swamy, and Rico Zenklusen. Approximate multi-matroid intersection via iterative refinement. Math. Program., 183(1):397-418, 2020. Google Scholar
  20. Konstantin Makarychev and Maxim Sviridenko. Solving optimization problems with diseconomies of scale via decoupling. J. ACM, 65(6):42:1-42:27, 2018. Google Scholar
  21. Albert W Marshall, Ingram Olkin, and Barry Arnold. Inequalities: Theory of Majorization and Its Applications. Springer series in statistics. Springer, 2011. Google Scholar
  22. Marco Molinaro. Stochastic 𝓁_p load balancing and moment problems via the L-function method. In Proceedings of the 30th SODA, pages 343-354, 2019. Google Scholar
  23. David B. Shmoys and Éva Tardos. An approximation algorithm for the generalized assignment problem. Math. Program., 62:461-474, 1993. Google Scholar
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