Scheduling to minimize total weighted completion time: Performance guarantees of LPbased heuristics and lower bounds
 Andreas S. Schulz
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Abstract
There has been recent success in using polyhedral formulations of scheduling problems not only to obtain good lower bounds in practice but also to develop provably good approximation algorithms. Most of these formulations rely on binary decision variables that are a kind of assignment variables. We present quite simple polynomialtime approximation algorithms that are based on linear programming formulations with completion time variables and give the best known performance guarantees for minimizing the total weighted completion time in several scheduling environments. This amplifies the importance of (appropriate) polyhedral formulations in the design of approximation algorithms with good worstcase performance guarantees.
In particular, for the problem of minimizing the total weighted completion time on a single machine subject to precedence constraints we present a polynomialtime approximation algorithm with performance ratio better than 2. This outperforms a (4 + ε)approximation algorithm very recently proposed by Hall, Shmoys, and Wein that is based on timeindexed formulations. A slightly extended formulation leads to a performance guarantee of 3 for the same problem but with release dates. This improves a factor of 5.83 for the same problem and even the 4approximation algorithm for the problem with release dates but without precedence constraints, both also due to Hall, Shmoys, and Wein.
By introducing new linear inequalities, we also show how to extend our technique to parallel machine problems. This leads, for instance, to the best known approximation algorithm for scheduling jobs with release dates on identical parallel machines. Finally, for the flow shop problem to minimize the total weighted completion time with both precedence constraints and release dates we present the first approximation algorithm that achieves a worstcase performance guarantee that is linear in the number of machines. We even extend this to multiprocessor flow shop scheduling.
The proofs of these results also imply guarantees for the lower bounds obtained by solving the proposed linear programming relaxations. This emphasizes the strength of linear programming formulations using completion time variables.
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 Title
 Scheduling to minimize total weighted completion time: Performance guarantees of LPbased heuristics and lower bounds
 Book Title
 Integer Programming and Combinatorial Optimization
 Book Subtitle
 5th International IPCO Conference Vancouver, British Columbia, Canada, June 3–5, 1996 Proceedings
 Pages
 pp 301315
 Copyright
 1996
 DOI
 10.1007/3540613102_23
 Print ISBN
 9783540613107
 Online ISBN
 9783540684534
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1084
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 Andreas S. Schulz ^{(1)}
 Author Affiliations

 1. Fachbereich Mathematik (MA 61), Technische Universität Berlin, Straße des 17. Juni 136, D10623, Berlin, Germany
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