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Mathematical Programming

, Volume 147, Issue 1–2, pp 1–23 | Cite as

An efficient polynomial time approximation scheme for load balancing on uniformly related machines

  • Leah EpsteinEmail author
  • Asaf Levin
Full Length Paper Series A

Abstract

We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, \(C_i\) denotes the completion time of machine i. Our goal is to find a schedule that minimizes or maximizes \(\sum _{i=1}^m C_i^p\) for a fixed value of p such that \(0<p<\infty \). For \(p>1\) the minimization problem is equivalent to the well-known problem of minimizing the \(\ell _p\) norm of the vector of the completion times of the machines, and for \(0<p<1\), the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS’s for this natural class of load balancing problems.

Keywords

EPTAS Load balancing Scheduling Approximation algorithms 

Mathematics Subject Classification

68W25 Approximation algorithms 68W40 Analysis of algorithms 68Q25 Analysis of algorithms and problem complexity 90C27 Combinatorial optimization  90C59 Approximation methods and heuristics 

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Faculty of Industrial Engineering and ManagementThe TechnionHaifaIsrael

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