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Additive Approximation Schemes for Load Balancing Problems

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

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

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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}.

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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

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

Funding

  • Rohwedder, Lars: Swiss National Science Foundation project 200021-184656.
  • Wiese, Andreas: Partially supported by FONDECYT Regular grant 1200173.

Acknowledgements

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

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