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

TL;DR

This paper presents the first efficient polynomial time approximation schemes for load balancing on uniformly related machines, optimizing the sum of completion times raised to a power p, using a novel shifting technique.

Contribution

Introduces a new EPTAS for load balancing on uniformly related machines, employing a non-standard shifting technique and dynamic programming.

Findings

First EPTAS for these load balancing problems

Effective use of shifting technique with forbidden work intervals

Partitioning into sub-problems enables polynomial-time approximation

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 which 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 forbidden work. 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.