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Linear, Exponential, but Nothing Else

On Pure Nash Equilibria in Congestion Games and Priority Rules for Single-Machine Scheduling

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Abstract

We consider two seemingly unrelated resource allocation problems and show that they share a deep structural property. In the first problem, we schedule jobs on a single machine to minimize the sum of the jobs’ cost where each job’s cost is determined by a job-specific function of its completion time. In the second problem, we consider weighted congestion games and are interested in the existence of pure Nash equilibria. We show that the classes of delay cost functions for which the scheduling problem admits a priority rule are exactly the classes of resource cost functions that guarantee the existence of a pure Nash equilibrium in weighted congestion games. These classes of cost functions are those that contain only linear functions or exponential functions, but not both.

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Acknowledgments

Part of the results of this chapter are joint work with Tobias Harks and Rolf H. Möhring. I also wish to thank Wiebke Höhn for introducing me to the topic of priority rules for single-machine scheduling problems.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Max Klimm

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  1. Max Klimm
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Correspondence to Max Klimm .

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Editors and Affiliations

  1. Technische Universität München, Munich, Germany

    Andreas S. Schulz

  2. Institute of Mathematics, Technische Universität Berlin, Berlin, Berlin, Germany

    Martin Skutella

  3. Technische Universität Braunschweig, Braunschweig, Germany

    Sebastian Stiller

  4. Institute of Theoretical Informatics, Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany

    Dorothea Wagner

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Klimm, M. (2015). Linear, Exponential, but Nothing Else. In: Schulz, A., Skutella, M., Stiller, S., Wagner, D. (eds) Gems of Combinatorial Optimization and Graph Algorithms . Springer, Cham. https://doi.org/10.1007/978-3-319-24971-1_11

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