, Volume 62, Issue 1, pp 461474
First online:
An approximation algorithm for the generalized assignment problem
 David B. ShmoysAffiliated withSchool of Operations Research and Engineering, Cornell University
 , Éva TardosAffiliated withSchool of Operations Research and Engineering, Cornell University
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The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timep _{ ij } and incurs a cost ofc _{ ij }; each machinei is available forT _{ i } time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomialtime algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2T _{ i } time units.
We also extend this result to a variant of the problem where, instead of a fixed processing timep _{ ij }, there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomialtime 2approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomialtime algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T.
Key words
Approximation algorithms generalized assignment problem scheduling unrelated parallel machines Title
 An approximation algorithm for the generalized assignment problem
 Journal

Mathematical Programming
Volume 62, Issue 13 , pp 461474
 Cover Date
 199302
 DOI
 10.1007/BF01585178
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Approximation algorithms
 generalized assignment problem
 scheduling unrelated parallel machines
 Industry Sectors
 Authors

 David B. Shmoys ^{(1)}
 Éva Tardos ^{(1)}
 Author Affiliations

 1. School of Operations Research and Engineering, Cornell University, 232 ET&C Building, 14853, Ithaca, NY, USA