Summary: Scheduling Parallel Jobs with Linear Speedup
Alexander Grigoriev and Marc Uetz
Maastricht University, Quantitative Economics, P.O.Box 616,
6200 MD Maastricht, The Netherlands.
Abstract We consider a scheduling problem where a set of jobs is a-
priori distributed over parallel machines. The processing time of any job
is dependent on the usage of a scarce renewable resource, e.g. personnel.
An amount of k units of that resource can be allocated to the jobs at
any time, and the more of that resource is allocated to a job, the smaller
its processing time. The dependence of processing times on the amount
of resources is linear for any job. The objective is to find a resource allo-
cation and a schedule that minimizes the makespan. Utilizing an integer
quadratic programming relaxation, we show how to obtain a (3 + )-
approximation algorithm for that problem, for any > 0. This gener-
alizes and improves previous results, respectively. Our approach relies
on a fully polynomial time approximation scheme to solve the quadratic
programming relaxation. This result is interesting in itself, because the
underlying quadratic program is NP-hard to solve. We also derive lower
bounds, and discuss further generalizations of the results.