, Volume 46, Issue 1, pp 259271
First online:
Approximation algorithms for scheduling unrelated parallel machines
 Jan Karel LenstraAffiliated withEindhoven University of TechnologyCentre for Mathematics and Computer Science
 , David B. ShmoysAffiliated withCornell University
 , Éva TardosAffiliated withCornell University
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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 worstcase 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 Title
 Approximation algorithms for scheduling unrelated parallel machines
 Journal

Mathematical Programming
Volume 46, Issue 13 , pp 259271
 Cover Date
 199001
 DOI
 10.1007/BF01585745
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Scheduling
 parallel machines
 approximation algorithm
 worst case analysis
 linear programming
 integer programming
 rounding
 Industry Sectors
 Authors

 Jan Karel Lenstra ^{(1)} ^{(2)}
 David B. Shmoys ^{(3)}
 Éva Tardos ^{(3)}
 Author Affiliations

 1. Eindhoven University of Technology, Eindhoven, The Netherlands
 2. Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
 3. Cornell University, Ithaca, NY, USA