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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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References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice hall, Upper Saddle River (1993)

    MATH  Google Scholar 

  2. Aziz, H., Sørensen, T.B.: Path coalitional games (2011). arXiv preprint arXiv:1103.3310

  3. Baïou, M., Barahona, F.: On the nucleolus of shortest path games. In: Bilò, V., Flammini, M. (eds.) Algorithmic Game Theory, pp. 55–66. Springer International Publishing, Cham (2017)

    Chapter  Google Scholar 

  4. Chvatal, V.: Linear Programming. Macmillan, New York (1983)

    MATH  Google Scholar 

  5. Deng, X., Fang, Q., Sun, X.: Finding nucleolus of flow game. J. Comb. Optim. 18(1), 64–86 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Elkind, E., Pasechnik, D.: Computing the nucleolus of weighted voting games. In: Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 327–335. Society for Industrial and Applied Mathematics (2009)

  7. Faigle, U., Kern, W., Kuipers, J.: Note computing the nucleolus of min-cost spanning tree games is np-hard. Int. J. Game Theory 27(3), 443–450 (1998)

    Article  MATH  Google Scholar 

  8. Fang, Q., Li, B., Shan, X., Sun, X.: The least-core and nucleolus of path cooperative games. In: International Computing and Combinatorics Conference, pp. 70–82. Springer (2015)

  9. Fragnelli, V., Garcia-Jurado, I., Mendez-Naya, L.: On shortest path games. Math. Methods Oper. Res. 52(2), 251–264 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gillies, D.B.: Solutions to general non-zero-sum games. Contrib. Theory Games 4(40), 47–85 (1959)

    MathSciNet  MATH  Google Scholar 

  11. Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by canceling negative cycles. J. ACM (JACM) 36(4), 873–886 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Granot, D., Granot, F., Zhu, W.R.: Characterization sets for the nucleolus. Int. J. Game Theory 27(3), 359–374 (1998). https://doi.org/10.1007/s001820050078

    Article  MathSciNet  MATH  Google Scholar 

  13. Hartmann, M., Orlin, J.B.: Finding minimum cost to time ratio cycles with small integral transit times. Networks 23(6), 567–574 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huberman, G.: The nucleolus and the essential coalitions. In: Bensoussan, A., Lions, J.L. (eds.) Analysis and Optimization of Systems, pp. 416–422. Springer, Berlin (1980)

    Chapter  Google Scholar 

  15. Kalai, E., Zemel, E.: Generalized network problems yielding totally balanced games. Oper. Res. 30(5), 998–1008 (1982)

    Article  MATH  Google Scholar 

  16. Kern, W., Paulusma, D.: Matching games: the least core and the nucleolus. Math. Oper. Res. 28(2), 294–308 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kopelowitz, A.: Computation of the kernels of simple games and the nucleolus of n-person games. Technical Report, DTIC Document (1967)

  18. Megiddo, N.: Computational complexity of the game theory approach to cost allocation for a tree. Math. Oper. Res. 3(3), 189–196 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  19. Potters, J., Reijnierse, H., Biswas, A.: The nucleolus of balanced simple flow networks. Games Econ. Behav. 54(1), 205–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17(6), 1163–1170 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. Solymosi, T., Raghavan, T.E.: An algorithm for finding the nucleolus of assignment games. Int. J. Game Theory 23(2), 119–143 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Solymosi, T., Sziklai, B.: Characterization sets for the nucleolus in balanced games. Oper. Res. Lett. 44(4), 520–524 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sziklai, B., Fleiner, T., Solymosi, T.: On the core and nucleolus of directed acyclic graph games. Math. Program. 163(1), 243–271 (2017). https://doi.org/10.1007/s10107-016-1062-y

    Article  MathSciNet  MATH  Google Scholar 

  24. Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34(2), 250–256 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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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