A deterministic subexponential algorithm for solving parity games

The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matoušek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. Our deterministic algorithm is almost as fast as the randomized algorithms mentioned above.

[1]  Henrik Björklund,et al.  Randomized Subexponential Algorithms for Infinite Games , 2004 .

[2]  Stephan Merz,et al.  Model Checking , 2000 .

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

[4]  Wieslaw Zielonka,et al.  Infinite Games on Finitely Coloured Graphs with Applications to Automata on Infinite Trees , 1998, Theor. Comput. Sci..

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

[6]  N. Halman,et al.  Discrete and Lexicographic Helly Theorems and Their Relations to LP-type problems , 2004 .

[7]  Marcin Jurdzinski,et al.  Small Progress Measures for Solving Parity Games , 2000, STACS.

[8]  Mihalis Yannakakis,et al.  How easy is local search? , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

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

[10]  Bernd Gärtner.  The Random-Facet simplex algorithm on combinatorial cubes , 2002, Random Struct. Algorithms.

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

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

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

[14]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[15]  Sven Schewe.  Solving Parity Games in Big Steps , 2007, FSTTCS.

[16]  Bernd Gärtner,et al.  Simple Stochastic Games and P-Matrix Generalized Linear Complementarity Problems , 2005, FCT.

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

[18]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

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

[20]  Henrik Björklund,et al.  A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games , 2007, Discret. Appl. Math..

[21]  M. Paterson,et al.  A deterministic subexponential algorithm for solving parity games , 2006, SODA 2006.

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

[23]  Ola Svensson,et al.  Linear Complementarity and P-Matrices for Stochastic Games , 2006, Ershov Memorial Conference.

[24]  Rahul Savani,et al.  A Simple P-Matrix Linear Complementarity Problem for Discounted Games , 2008, CiE.

[25]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[26]  Rupak Majumdar,et al.  Quantitative solution of omega-regular games , 2004, J. Comput. Syst. Sci..

[27]  Orna Kupferman,et al.  On the Complexity of Parity Word Automata , 2001, FoSSaCS.

[28]  Jan Obdrzálek,et al.  Fast Mu-Calculus Model Checking when Tree-Width Is Bounded , 2003, CAV.

[29]  Brendan A. Juba.  On the Hardness of Simple Stochastic Games , 2005 .

[30]  Gil Kalai,et al.  A subexponential randomized simplex algorithm (extended abstract) , 1992, STOC '92.

[31]  Thomas Wilke,et al.  Automata logics, and infinite games: a guide to current research , 2002 .

[32]  Marcin Jurdziński,et al.  Deciding the Winner in Parity Games is in UP \cap co-Up , 1998, Inf. Process. Lett..

[33]  Robert McNaughton,et al.  Infinite Games Played on Finite Graphs , 1993, Ann. Pure Appl. Logic.

[34]  Krishnendu Chatterjee,et al.  Quantitative stochastic parity games , 2004, SODA '04.

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