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Approximating Precedence-Constrained Single Machine Scheduling by Coloring

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2006, RANDOM 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4110))

Abstract

This paper investigates the relationship between the dimension theory of partial orders and the problem of scheduling precedence-constrained jobs on a single machine to minimize the weighted completion time. Surprisingly, we show that the vertex cover graph associated to the scheduling problem is exactly the graph of incomparable pairs defined in dimension theory. This equivalence gives new insights on the structure of the problem and allows us to benefit from known results in dimension theory. In particular, the vertex cover graph associated to the scheduling problem can be colored efficiently with at most k colors whenever the associated poset admits a polynomial time computable k-realizer. Based on this approach, we derive new and better approximation algorithms for special classes of precedence constraints, including convex bipartite and semi-orders, for which we give \((1+\frac{1}{3})\)-approximation algorithms. Our technique also generalizes to a richer class of posets obtained by lexicographic sum.

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Author information

Authors and Affiliations

  1. University of Liverpool, Great Britain

    Christoph Ambühl

  2. IDSIA, Switzerland

    Monaldo Mastrolilli & Ola Svensson

Authors
  1. Christoph Ambühl
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  2. Monaldo Mastrolilli
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  3. Ola Svensson
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Editor information

Editors and Affiliations

  1. Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord - Ed. Omega, 240 Jordi Girona Salgado, 1-3 E-08034, Barcelona

    Josep Díaz

  2. Institute for Computer Science, University of Kiel, Olshausenstrasse 40, 24118, Kiel, Germany

    Klaus Jansen

  3. Centre Universitaire d’Informatique, Battelle Bâtiment A, Route de Drize 7,, 1227, Carouge, Geneva, Switzerland

    José D. P. Rolim

  4. School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Uri Zwick

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© 2006 Springer-Verlag Berlin Heidelberg

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Ambühl, C., Mastrolilli, M., Svensson, O. (2006). Approximating Precedence-Constrained Single Machine Scheduling by Coloring. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006 2006. Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11830924_4

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  • DOI: https://doi.org/10.1007/11830924_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38044-3

  • Online ISBN: 978-3-540-38045-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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