Mathematical Programming

, Volume 141, Issue 1, pp 193–215

Computing pure Nash and strong equilibria in bottleneck congestion games

  • Tobias Harks
  • Martin Hoefer
  • Max Klimm
  • Alexander Skopalik
Open AccessFull Length Paper Series A

DOI: 10.1007/s10107-012-0521-3

Cite this article as:
Harks, T., Hoefer, M., Klimm, M. et al. Math. Program. (2013) 141: 193. doi:10.1007/s10107-012-0521-3


Bottleneck congestion games properly model the properties of many real-world network routing applications. They are known to possess strong equilibria—a strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or single-commodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and single-commodity games is strongly NP-hard. In addition, we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics. They continue to hold even for convergence to approximate equilibria.


Bottleneck congestion gamesComputation of strong equilibriaImprovement dynamics

Mathematics Subject Classification

91A10 Noncooperative games91A46 Combinatorial games91-08 Computational methods90B18 Communication networks
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© The Author(s) 2012

Authors and Affiliations

  • Tobias Harks
    • 1
  • Martin Hoefer
    • 2
  • Max Klimm
    • 3
  • Alexander Skopalik
    • 4
  1. 1.School of Business and EconomicsMaastricht UniversityMaastrichtThe Netherlands
  2. 2.Department of Computer ScienceRWTH Aachen UniversityAachenGermany
  3. 3.Department of MathematicsTU BerlinBerlinGermany
  4. 4.TU DortmundDortmundGermany