Single Machine Scheduling with Precedence Constraints
 José R. Correa,
 Andreas S. Schulz
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Abstract
We discuss the problem of sequencing precedenceconstrained jobs on a single machine to minimize the average weighted completion time. This problem has attracted much attention in the mathematical programming community since Sidney’s pioneering work in 1975. We look at the problem from a polyhedral perspective and uncover a relation between Sidney’s decomposition theorem and different linear programming relaxations. More specifically, we present a generalization of Sidney’s result, which particularly allows us to reason that virtually all known 2approximation algorithms comply with his decomposition. Moreover, we establish a connection between the singlemachine scheduling problem and the vertex cover problem. Indeed, in the special case of seriesparallel precedence constraints, we prove that the sequencing problem can be seen as a special case of vertex cover. We also argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem. Finally, we provide a characterization of the active inequalities of a linear programming relaxation in completion time variables.
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 Title
 Single Machine Scheduling with Precedence Constraints
 Book Title
 Integer Programming and Combinatorial Optimization
 Book Subtitle
 10th International IPCO Conference, New York, NY, USA, June 711, 2004. Proceedings
 Pages
 pp 283297
 Copyright
 2004
 DOI
 10.1007/9783540259602_22
 Print ISBN
 9783540221135
 Online ISBN
 9783540259602
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 3064
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 Additional Links
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 Industry Sectors
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 Editors

 Daniel Bienstock ^{(14)}
 George Nemhauser ^{(15)}
 Editor Affiliations

 14. Department of IEOR, Columbia University
 15. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
 Authors

 José R. Correa ^{(16)}
 Andreas S. Schulz ^{(16)}
 Author Affiliations

 16. Operations Research Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 021394307, USA
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