Approximation algorithms for scheduling unrelated parallel machines Jan Karel Lenstra David B. Shmoys Éva Tardos Article Received: 01 October 1987 Revised: 26 August 1988 DOI :
10.1007/BF01585745

Cite this article as: Lenstra, J.K., Shmoys, D.B. & Tardos, É. Mathematical Programming (1990) 46: 259. doi:10.1007/BF01585745
Abstract We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep _{ij} . The objective is to find a schedule that minimizes the makespan.

Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints.

In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

Key words Scheduling parallel machines approximation algorithm worst case analysis linear programming integer programming rounding A preliminary version of this paper appeared in theProceedings of the 28th Annual IEEE Symposium on the Foundations of Computer Science (Computer Society Press of the IEEE, Washington, D.C., 1987) pp. 217–224.

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Google Scholar Authors and Affiliations Jan Karel Lenstra David B. Shmoys Éva Tardos 1. Eindhoven University of Technology Eindhoven The Netherlands 2. Centre for Mathematics and Computer Science Amsterdam The Netherlands 3. Cornell University Ithaca USA