A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games

We suggest the first strongly subexponential and purely combinatorial algorithm for mean payoff games. It is based on solving a new controlled version of the shortest paths problem. By selecting exactly one outgoing edge in each of the controlled vertices we want to maximize the shortest distances to the unique sink. Mean payoff games easily reduce to this problem. To compute the longest shortest paths, player MAX selects a strategy (one edge in each controlled vertex) and player MIN responds by evaluating shortest paths to the sink in the remaining graph. Then MAX locally changes choices in controlled vertices, making attractive switches that seem to increase shortest paths (under the current evaluation). We show that this is a monotonic strategy improvement, and every locally optimal strategy is globally optimal. A careful choice of the next iterate results in a randomized algorithm of complexity min(poly(n).W, 2 o( √ n log n) ), which is simultaneously pseudopolynomial (W is the maximal absolute edge weight) and subexponential in the number of vertices n. All previous algorithms for mean payoff games were either exponential or pseudopolynomial (which is purely exponential for exponentially large edge weights).

[1]  Henrik Björklund,et al.  Memoryless determinacy of parity and mean payoff games: a simple proof , 2004, Theor. Comput. Sci..

[2]  Anne Condon,et al.  On Algorithms for Simple Stochastic Games , 1990, Advances In Computational Complexity Theory.

[3]  R. Karp,et al.  On Nonterminating Stochastic Games , 1966 .

[4]  E. Allen Emerson,et al.  Model Checking and the Mu-calculus , 1996, Descriptive Complexity and Finite Models.

[5]  Nicolai N. Pisaruk,et al.  Mean Cost Cyclical Games , 1999, Math. Oper. Res..

[6]  E. Allen Emerson,et al.  Tree automata, mu-calculus and determinacy , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[7]  H. Moulin.  Extensions of two person zero sum games , 1976 .

[8]  A. Ehrenfeucht,et al.  Positional strategies for mean payoff games , 1979 .

[9]  A. Karzanov,et al.  Cyclic games and an algorithm to find minimax cycle means in directed graphs , 1990 .

[10]  Kathy Williamson Hoke,et al.  Completely unimodal numberings of a simple polytope , 1988, Discret. Appl. Math..

[11]  A. Prasad Sistla,et al.  On Model-Checking for Fragments of µ-Calculus , 1993, CAV.

[12]  Henrik Björklund,et al.  A Discrete Subexponential Algorithm for Parity Games , 2003, STACS.

[13]  Walter Ludwig,et al.  A Subexponential Randomized Algorithm for the Simple Stochastic Game Problem , 1995, Inf. Comput..

[14]  A. Puri.  Theory of hybrid systems and discrete event systems , 1996 .

[15]  Uri Zwick,et al.  The Complexity of Mean Payoff Games on Graphs , 1996, Theor. Comput. Sci..

[16]  Henrik Björklund,et al.  Complexity of Model Checking by Iterative Improvement: The Pseudo-Boolean Framework , 2003, Ershov Memorial Conference.

[17]  Anne Condon,et al.  The Complexity of Stochastic Games , 1992, Inf. Comput..

[18]  Micha Sharir,et al.  A subexponential bound for linear programming , 1992, SCG '92.

[19]  Marcin Jurdzinski,et al.  A Discrete Strategy Improvement Algorithm for Solving Parity Games , 2000, CAV.

[20]  Henrik Björklund,et al.  On Combinatorial Structure and Algorithms for Parity Games , 2003 .