Journal of Mathematical Sciences

, Volume 145, Issue 3, pp 4967–4974

Potential theory for mean payoff games

  • Y. M. Lifshits
  • D. S. Pavlov
Article

Abstract

The paper presents an O(mn2nlog Z) deterministic algorithm for solving the mean payoff game problem, m and n being the numbers of arcs and vertices, respectively, in the game graph, and Z being the maximum weight (the weights are assumed to be integers). The theoretical basis for the algorithm is the potential theory for mean payoff games. This theory allows one to restate the problem in terms of solving systems of algebraic equations with minima and maxima. Also, in order to solve the mean payoff game problem, the arc reweighting technique is used. To this end, simple modifications, which do not change the set of winning strategies, are applied to the game graph; in the end, a trivial instance of the problem is obtained. It is shown that any game graph can be simplified by n reweightings. Bibliography: 16 titles.

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References

  1. 1.
    H. Björklund, S. Sandberg, and S. Vorobyov, “Memoryless determinacy of parity and mean payoff games: a simple proof,” Theor. Comp. Sci., 310, 365–378 (2004).MATHCrossRefGoogle Scholar
  2. 2.
    H. Björklund, S. Sandberg, and S. Vorobyov, “A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games,” DIMACS Tech. Rep., No. 5 (2004).Google Scholar
  3. 3.
    Y. J. Chu and T. H. Liu, “On the shortest arborescence of a directed graph,” Sci. Sin., 14, 1396–1400 (1965).MATHGoogle Scholar
  4. 4.
    A. Ehrenfeucht and J. Mycielski, “Positional strategies for mean payoff games,” Int. J. Game Theory, 8, 109–113 (1979).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    V. A. Gurvich, A. V. Karzanov, and L. G. Khachiyan, “Cyclic games and an algorithm to find minimax cycle means in directed graphs,” Zh. Vychisl. Mat. Mat. Fiz., 28, 85–91 (1988).MATHMathSciNetGoogle Scholar
  6. 6.
    T. Gallai, “Maximum-Minimum Sätze über Graphen,” Acta Math. Acad. Sci. Hung., 9, 395–434 (1958).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Springer-Verlag, Berlin (2003).MATHGoogle Scholar
  8. 8.
    D. B. Johnson, “Efficient algorithms for shortest paths in sparse networks,” J. Assoc. Comp. Mach., 24, 1–13 (1977).MATHGoogle Scholar
  9. 9.
    M. Jurdziński, “Small progress measures for solving parity games,” in: Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science, H. Reichel and S. Tison, eds. (Lect. Notes Comp. Sci., 1770), Springer-Verlag, 2000, pp. 290–301.Google Scholar
  10. 10.
    M. Jurdziński, M. Paterson, and U. Zwick, “A deterministic subexponential algorithm for solving parity games,” in: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (2006), pp. 117–123.Google Scholar
  11. 11.
    H. Klauck, “Algorithms for parity games,” in: Automata, Logics, and Infinite Games, E. Grädel et al., eds. (Lect. Notes Comp. Sci., 2500), Springer-Verlag, 2002, pp. 107–129.Google Scholar
  12. 12.
    D. Kozen, “Results on the propositional µ-calculus,” Theor. Comp. Sci., 27, 333–354 (1983).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Logistics Quart., 2, 83–97 (1955).CrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Kwek and K. Mehlhorn, “Optimal search for rationals,” Inform. Proces. Letters, 86, 23–26 (2003).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    N. Pisaruk, “Mean cost cyclical games,” Math. Oper. Res., 24, 817–828 (1999).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    U. Zwick and M. Paterson, “The complexity of mean payoff games on graphs,” Theor. Comp. Sci., 158, 343–359 (1996).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Y. M. Lifshits
    • 1
  • D. S. Pavlov
    • 2
  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia
  2. 2.Department of MathematicsInstitute of Fine Mechanics and OpticsSt. PetersburgRussia

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