Abstract
We study the problem of minimizing the total weighted tardiness when scheduling unti-length jobs on a single machine, in the presence of large sets of identical jobs. Previously known algorithms, which do not exploit the set structure, are at best pseudo-polynomial, and may be prohibitively inefficient when the set sizes are large. We give a polynomial algorithm for the problem, whose number of operations is independent of the set sizes. The problem is reformulated as an integer program with a quadratic, non-separable objective and transportation constraints. Employing methods of real analysis, we prove a tight proximity result between the integer solution to that problem and a fractional solution of a related problem. The related problem is shown to be polynomially solvable, and a rounding algorithm applied to its solution gives the optimal integer solution to the original problem.
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Supported in part by the National Science Foundation under grant ECS-85-01988, and by the Office of Naval Research under grant N00014-88-K-0377.
Supported in part by Allon Fellowship, by Air Force grants 89-0512 and 90-0008 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center—NSF-STC88-09648. Part of the research of this author was performed in DIMACS Center, Rutgers University.
Supported in part by Air Force grant 84-0205.
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Hochbaum, D.S., Shamir, R. & Shanthikumar, J.G. A polynomial algorithm for an integer quadratic non-separable transportation problem. Mathematical Programming 55, 359–371 (1992). https://doi.org/10.1007/BF01581207
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DOI: https://doi.org/10.1007/BF01581207