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

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 Ω(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 ε> 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.

The work is fully supported by the National Natural Science Foundation of China [No.10861012,10561009], Natural Science Foundation of Yunnan Province [No.2006F0016M] and Foundation of Younger Scholar in Science and Technology of Yunnan Province [No.2007PY01-21].