Polynomial Approximation Schemes for the Max-Min Allocation Problem under a Grade of Service Provision

Authors:
Jianping Li;Weidong Li;Jianbo Li
Affiliations:
Department of Mathematics, Yunnan University, Kunming, China 650091;Department of Mathematics, Yunnan University, Kunming, China 650091;School of Management and Economics, Kunming Uni. of Sci. and Tech., China
Venue:
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
Year:
2009

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Abstract

The max-min allocation problem under a grade of service provision is defined in the following model: given a set ${\cal M}$ of m parallel machines and a set ${\cal J}$ of n jobs, where machines and jobs are all entitled to different levels of grade of service (GoS), each job $J_j\in {\cal J}$ has its processing time p j and it is only allocated to a machine M i whose GoS level is no more than the GoS level the job J j has. The goal is to allocate all jobs to m machines to maximize the minimum machine load , where the machine load of machine M i is the sum of the precessing times of jobs executed on M i . The best approximation algorithm [4] to solve this problem produces an allocation in which the minimum machine completion time is at least *** (logloglogm /loglogm ) of the optimal value. In this paper, we respectively present four approximation schemes to solve this problem and its two special versions: (1) a polynomial time approximation scheme (PTAS) with running time $O(mn^{O(1/\epsilon^2)})$ for the general version, where *** 0; (2) a PTAS and an fully polynomial time approximation scheme (FPTAS) with running time O (n ) for the version where the number m of machines is fixed; (3) a PTAS with running time O (n ) for the version where the number of GoS levels is bounded by k .