Mathematical Programming

, Volume 46, Issue 1–3, pp 259–271 | Cite as

Approximation algorithms for scheduling unrelated parallel machines

  • Jan Karel Lenstra
  • David B. Shmoys
  • Éva Tardos


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 


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Copyright information

© North-Holland 1990

Authors and Affiliations

  • Jan Karel Lenstra
    • 1
    • 2
  • David B. Shmoys
    • 3
  • Éva Tardos
    • 3
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands
  3. 3.Cornell UniversityIthacaUSA

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