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An Algorithm to Compute the Nucleolus of Shortest Path Games

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Abstract

We study a type of cooperative games introduced in Fragnelli et al. (Math Methods Oper Res 52(2):251–264, 2000) called shortest path games. They arise on a network that has two special nodes s and t. A coalition corresponds to a set of arcs and it receives a reward if it can connect s and t. A coalition also incurs a cost for each arc that it uses to connect s and t, thus the coalition must choose a path of minimum cost among all the arcs that it controls. These games are relevant to logistics, communication, or supply-chain networks. We give a polynomial combinatorial algorithm to compute the nucleolus. This vector reflects the relative importance of each arc to ensure the connectivity between s and t.

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Acknowledgements

We are grateful to an anonymous referee for his careful reading. His comments helped us to improve the presentation. This work has been partially supported by the French government research program “Investissements d’Avenir” through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) and the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01).

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Authors and Affiliations

  1. CNRS and Université Clermont Auvergne, Campus des Cézeaux, 1 Rue de la Chebarde, 63178, Aubière Cedex, France

    Mourad Baïou

  2. IBM T. J. Watson research Center, Yorktown Heights, NY, 10589, USA

    Francisco Barahona

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  1. Mourad Baïou
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  2. Francisco Barahona
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Correspondence to Francisco Barahona.

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Baïou, M., Barahona, F. An Algorithm to Compute the Nucleolus of Shortest Path Games. Algorithmica 81, 3099–3113 (2019). https://doi.org/10.1007/s00453-019-00574-9

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