Approximation algorithms for scheduling unrelated parallel machines

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.

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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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Lenstra, J.K., Shmoys, D.B. & Tardos, É. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 46, 259–271 (1990). https://doi.org/10.1007/BF01585745

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Key words

  • Scheduling
  • parallel machines
  • approximation algorithm
  • worst case analysis
  • linear programming
  • integer programming
  • rounding