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Encouraging Cooperation in Sharing Supermodular Costs

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2007, RANDOM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4627))

Abstract

We study the computational complexity and algorithmic aspects of computing the least core value of supermodular cost cooperative games, and uncover some structural properties of the least core of these games. We provide motivation for studying these games by showing that a particular class of optimization problems has supermodular optimal costs. This class includes a variety of problems in combinatorial optimization, especially in machine scheduling. We show that computing the least core value of supermodular cost cooperative games is NP-hard, and design approximation algorithms based on oracles that approximately determine maximally violated constraints. We apply our results to schedule planning games, or cooperative games where the costs arise from the minimum sum of weighted completion times on a single machine. By improving upon some of the results for general supermodular cost cooperative games, we are able to give an explicit formula for an element of the least core of schedule planning games, and design a fully polynomial time approximation scheme for computing the least core value of these games.

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Author information

Authors and Affiliations

  1. Sloan School of Management, Massachusetts Institute of Technology, 77 Massachusetts Avenue, E53-361, Cambridge, MA 02139, USA

    Andreas S. Schulz

  2. Operations Research Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, E40-130, Cambridge, MA 02139, USA

    Nelson A. Uhan

Authors
  1. Andreas S. Schulz
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  2. Nelson A. Uhan
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Editor information

Editors and Affiliations

  1. Computer Science Dept., Princeton University, 35 Olden Street, NJ 08540-5233, Princeton, USA

    Moses Charikar

  2. Institut für Informatik, Christian-Albrechts-Universität zu Kiel,, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Omer Reingold

  4. Centre Universitaire d’Informatique,, Battelle bâtiment A, route de Drize 7, 1227, Geneva, Carouge, Switzerland

    José D. P. Rolim

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© 2007 Springer-Verlag Berlin Heidelberg

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Schulz, A.S., Uhan, N.A. (2007). Encouraging Cooperation in Sharing Supermodular Costs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_20

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  • DOI: https://doi.org/10.1007/978-3-540-74208-1_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74207-4

  • Online ISBN: 978-3-540-74208-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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