The Parallel Complexity of Coloring Games

  • Guillaume Ducoffe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)


We wish to motivate the problem of finding decentralized lower-bounds on the complexity of computing a Nash equilibrium in graph games. While the centralized computation of an equilibrium in polynomial time is generally perceived as a positive result, this does not reflect well the reality of some applications where the game serves to implement distributed resource allocation algorithms, or to model the social choices of users with limited memory and computing power. As a case study, we investigate on the parallel complexity of a game-theoretic variation of graph coloring. These “coloring games” were shown to capture key properties of the more general welfare games and Hedonic games. On the positive side, it can be computed a Nash equilibrium in polynomial-time for any such game with a local search algorithm. However, the algorithm is time-consuming and it requires polynomial space. The latter questions the use of coloring games in the modeling of information-propagation in social networks. We prove that the problem of computing a Nash equilibrium in a given coloring game is PTIME-hard, and so, it is unlikely that one can be computed with an efficient distributed algorithm. The latter brings more insights on the complexity of these games.


Nash Equilibrium Parallel Machine Colour Class Parallel Complexity Proper Coloring 
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  1. 1.
    Ballester, C.: NP-completeness in hedonic games. Games Econ. Behav. 49(1), 1–30 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics. Springer, London (2008)zbMATHCrossRefGoogle Scholar
  3. 3.
    Burani, N., Zwicker, W.S.: Coalition formation games with separable preferences. Math. Soc. Sci. 45(1), 27–52 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chatzigiannakis, I., Koninis, C., Panagopoulou, P.N., Spirakis, P.G.: Distributed game-theoretic vertex coloring. In: OPODIS 2010, pp. 103–118 (2010)Google Scholar
  5. 5.
    Condon, A.: A theory of strict P-completeness. Comput. Complex. 4(3), 220–241 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ducoffe, G., Mazauric, D., Chaintreau, A.: The complexity of hedonic coalitions under bounded cooperation (submitted)Google Scholar
  7. 7.
    Feigenbaum, J., Papadimitriou, C.H., Shenker, S.: Sharing the cost of multicast transmissions. J. Comput. Syst. Sci. 63(1), 21–41 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gairing, M., Savani, R.: Computing stable outcomes in hedonic games. In: SAGT 2010, pp. 174–185 (2010)Google Scholar
  9. 9.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  10. 10.
    Karloff, H., Suri, S., Vassilvitskii, S.: A model of computation for MapReduce. In: SODA 2010, pp. 938–948 (2010)Google Scholar
  11. 11.
    Kearns, M., Suri, S., Montfort, N.: An experimental study of the coloring problem on human subject networks. Science 313(5788), 824–827 (2006)CrossRefGoogle Scholar
  12. 12.
    Kleinberg, J., Ligett, K.: Information-sharing in social networks. Games Econ. Behav. 82, 702–716 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Marden, J.R., Wierman, A.: Distributed welfare games. Oper. Res. 61(1), 155–168 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Panagopoulou, P.N., Spirakis, P.G.: A game theoretic approach for efficient graph coloring. In: ISAAC 2008, pp. 183–195 (2008)Google Scholar
  15. 15.
    Papadimitriou, C.H.: Computational Complexity. Wiley, Reading (2003)zbMATHGoogle Scholar
  16. 16.
    Vassilevska Williams, V.: Fine-Grained algorithms and complexity (invited talk). In: STACS 2016 (2016)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Université Côte D’Azur, Inria, CNRS, I3SSophia AntipolisFrance

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