Potential theory for mean payoff games Y. M. Lifshits D. S. Pavlov Article DOI :
10.1007/s10958-007-0331-y

Cite this article as: Lifshits, Y.M. & Pavlov, D.S. J Math Sci (2007) 145: 4967. doi:10.1007/s10958-007-0331-y
Abstract The paper presents an O(mn2^{n} log 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.

__________

Translated from Zapiski Nauchnykh Seminarov POMI , Vol. 340, 2006, pp. 61–75.

References 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).

MATH CrossRef Google Scholar 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).

3.

Y. J. Chu and T. H. Liu, “On the shortest arborescence of a directed graph,”

Sci. Sin. ,

14 , 1396–1400 (1965).

MATH Google Scholar 4.

A. Ehrenfeucht and J. Mycielski, “Positional strategies for mean payoff games,”

Int. J. Game Theory ,

8 , 109–113 (1979).

MATH CrossRef MathSciNet Google Scholar 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).

MATH MathSciNet Google Scholar 6.

T. Gallai, “Maximum-Minimum Sätze über Graphen,”

Acta Math. Acad. Sci. Hung. ,

9 , 395–434 (1958).

MATH CrossRef MathSciNet Google Scholar 7.

A. Schrijver,

Combinatorial Optimization: Polyhedra and Efficiency , Springer-Verlag, Berlin (2003).

MATH Google Scholar 8.

D. B. Johnson, “Efficient algorithms for shortest paths in sparse networks,”

J. Assoc. Comp. Mach. ,

24 , 1–13 (1977).

MATH Google Scholar 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.

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.

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.

12.

D. Kozen, “Results on the propositional µ-calculus,”

Theor. Comp. Sci. ,

27 , 333–354 (1983).

MATH CrossRef MathSciNet Google Scholar 13.

H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Logistics Quart.,

2 , 83–97 (1955).

CrossRef MathSciNet Google Scholar 14.

S. Kwek and K. Mehlhorn, “Optimal search for rationals,”

Inform. Proces. Letters ,

86 , 23–26 (2003).

CrossRef MATH MathSciNet Google Scholar 15.

N. Pisaruk, “Mean cost cyclical games,”

Math. Oper. Res. ,

24 , 817–828 (1999).

MATH CrossRef MathSciNet Google Scholar 16.

U. Zwick and M. Paterson, “The complexity of mean payoff games on graphs,”

Theor. Comp. Sci. ,

158 , 343–359 (1996).

MATH CrossRef MathSciNet Google Scholar © Springer Science+Business Media, Inc. 2007

Authors and Affiliations Y. M. Lifshits D. S. Pavlov 1. St. Petersburg Department of the Steklov Mathematical Institute St. Petersburg Russia 2. Department of Mathematics Institute of Fine Mechanics and Optics St. Petersburg Russia