# Approximation Schemes for Scheduling with Availability Constraints

## Abstract

We investigate the problems of scheduling *n* weighted jobs to *m* identical machines with availability constraints. We consider two different models of availability constraints: the preventive model where the unavailability is due to preventive machine maintenance, and the fixed job model where the unavailability is due to a priori assignment of some of the *n* jobs to certain machines at certain times. Both models have applications such as turnaround scheduling or overlay computing. In both models, the objective is to minimize the total weighted completion time. We assume that *m* is a constant, and the jobs are non-resumable. For the preventive model, it has been shown that there is no approximation algorithm if all machines have unavailable intervals even when *w* _{ i } = *p* _{ i } for all jobs. In this paper, we assume there is one machine permanently available and the processing time of each job is equal to its weight for all jobs. We develop the first PTAS when there are constant number of unavailable intervals. One main feature of our algorithm is that the classification of large and small jobs is with respect to each individual interval, thus not fixed. This classification allows us (1) to enumerate the assignments of large jobs efficiently; (2) and to move small jobs around without increasing the objective value too much, and thus derive our PTAS. Then we show that there is no FPTAS in this case unless *P* = *NP*.

For fixed job model, we first show that if job weights are arbitrary then there is no constant approximation for a single machine with 2 fixed jobs or for two machines with one fixed job on each machine, unless *P* = *NP* . As the preventive model, we assume that the weight of a job is the same as its processing time for all jobs. We show that the PTAS for the preventive model can be extended to solve this problem when the number of fixed jobs and the number of machines are both constants.

## Keywords

Schedule Problem Completion Time Single Machine Feasible Schedule Identical Machine## Preview

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