The load rebalancing problem

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Abstract

In the classical load balancing or multiprocessor scheduling problem, we are given a sequence of jobs of varying sizes and are asked to assign each job to one of the m empty processors. A typical objective is to minimize the makespan, which is the load on the heaviest loaded processor. Since in most real world scenarios the load is a dynamic measure, the initial assignment may not remain optimal over time. Motivated by such considerations in a variety of systems, we formulate the problem of load rebalancing—given a possibly suboptimal assignment of jobs to processors, relocate a set of the jobs so as to decrease the makespan. Specifically, the goal is to achieve the best possible makespan under the constraint that no more than k jobs are relocated. We also consider the weighted version of this problem where there is an arbitrary cost associated with each job's relocation. The problem is NP-hard and hence, we focus on approximation algorithms. We construct an algorithm which achieves a 1.5-approximation, with near linear running time. We also show that the problem has a PTAS, thereby resolving the complexity issue. Finally, we investigate the approximability of several extensions of the load rebalancing model.

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A preliminary version of this paper appeared in 15th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), 2003.

1

Supported in part by a Stanford Graduate Fellowship, NSF Grants EIA-0137761 and ITR-0331640, and a grant from SNRC.

2

Supported in part by NSF Grant IIS-0118173 and ITR-0331640, an Okawa Foundation Research Grant, an SNRC grant, and grants from Microsoft and Veritas.

3

Supported by a GRPW fellowship from Bell Labs, Lucent Technologies.

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