Combinatorial structure and randomized subexponential algorithms for infinite games

The complexity of solving infinite games, including parity, mean payoff, and simple stochastic, is an important open problem in verification, automata, and complexity theory. In this paper, we develop an abstract setting for studying and solving such games, based on function optimization over certain discrete structures. We introduce new classes of recursively local-global (RLG) and partial recursively local-global (PRLG) functions, and show that strategy evaluation functions for simple stochastic, mean payoff, and parity games belong to these classes.In this setting, we suggest randomized subexponential algorithms appropriate for RLG-and PRLG-function optimization. We show that the subexponential algorithms for combinatorial linear programming, due to Kalai and Matousek, Sharir, Welzl, can be adapted for optimizing the RLG-and PRLG-functions.

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