Scheduling Parallel Jobs with Linear Speedup

Abstract

We consider a scheduling problem where a set of jobs is apriori 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 allocation and a schedule that minimizes the makespan. Utilizing an integer quadratic programming relaxation, we show how to obtain a (3 + ε)-approximation algorithm for that problem, for any ε > 0. This generalizes 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.

Machine scheduling with resource dependent processing times

“…For that problem, they derive a (3 + ε)-approximation algorithm, and for the problem with m = 2 machines, they derive (weak) NPhardness and a fully polynomial time approximation scheme [8]. In a recent paper, Grigoriev and Uetz [9] have generalized the approximation result of [8]. The model of [9] is a dedicated machine setting as well, and assumes a linear time-resource tradeoff: There are k units of a renewable resource available, and the processing time p j of any job becomes p js =p j − b j s if s of the k resources are used.…”

“…In a recent paper, Grigoriev and Uetz [9] have generalized the approximation result of [8]. The model of [9] is a dedicated machine setting as well, and assumes a linear time-resource tradeoff: There are k units of a renewable resource available, and the processing time p j of any job becomes p js =p j − b j s if s of the k resources are used. Using quadratic programming relaxations, a (3 + ε)-approximation algorithm is derived in [9], for an arbitrary number of machines.…”

“…For the special case of identical parallel machines, we show in Section 6 how to improve the performance bound to (5 + ε), for any ε > 0. This approach is based upon another integer programming relaxation, and it lends some ideas from [9]. More precisely, we use a fully polynomial time approximation scheme (FPTAS), together with a sort of approximate binary search, to solve the integer programming relaxation with sufficiently large precision.…”

“…First, recall that x LP denotes the optimal fractional solution for the linear programming relaxation of (1)-(5), for C = C * . Using x LP = (x LP ijs ), defineỹ = (ỹ ij ) as in (6), and apply the rounding defined by (9). This yields a fractional solutionx = (x ijs ) that is nonzero only for one resource index s = s ij , for any pair of i and j, as defined in (8).…”

Scheduling jobs with time-resource tradeoff via nonlinear programming

“…We thank Gerhard Woeginger and Frits Spieksma for several helpful suggestions on a previous version of this paper [19], and the anonymous referees for their helpful comments and useful references.…”