 
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.
Email: {a.grigoriev,m.uetz}@ke.unimaas.nl
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 NPhard to solve. We also derive lower
bounds, and discuss further generalizations of the results.
