Approximation schemes for parallel machine scheduling with availability constraints

• Published in: Discrete Applied Mathematics Vol. 159; no. 15; pp. 1555 - 1565
• Main Authors: Fu, Bin, Huo, Yumei, Zhao, Hairong
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 06-09-2011; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Approximations and expansions; Availability; Classification; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Computer systems performance. Reliability; Exact sciences and technology; Inapproximability; Intervals; Mathematical analysis; Mathematical models; Mathematics; Parallel machine; PTAS; Scheduling; Sciences and techniques of general use; Software; Theoretical computing; Total weighted completion time; Parallel machine; Total weighted completion time; PTAS; Inapproximability; Availability; Computer theory; Constraint; Polynomial approximation; Parallel machines; Processing time; Scheduling; Approximation algorithm; Computing; Optimization; Polynomial time; Assignment; Completion; Classification; Combinatorics; Overlay; Single machine
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2011.06.007

We investigate the problems of scheduling
n
weighted jobs to
m
parallel machines with availability constraints. We consider two different models of availability constraints: the preventive model, in which the unavailability is due to preventive machine maintenance, and the fixed job model, in which 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 that 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 if
w
i
=
p
i
for all jobs. In this paper, we assume that there is one machine that is permanently available and that the processing time of each job is equal to its weight for all jobs. We develop the first polynomial-time approximation scheme (PTAS) when there is a 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, and thus not fixed. This classification allows us (1) to enumerate the assignments of large jobs efficiently; and (2) to move small jobs around without increasing the objective value too much, and thus derive our PTAS. Next, we show that there is no fully polynomial-time approximation scheme (FPTAS) in this case unless
P
=
N
P
.
For the fixed job model, it has been shown 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
=
N
P
. In this paper, 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.
► Minimizing the total weighted completion time with availability constraints is studied. ► The preventive model and the fixed job model are considered. ► For both models, we develop approximation schemes under some constraints.