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... of iterations when solved by policyOliver Friedmann 11 iteration. The lower bound result immediately transfers to mean payoff games, discounted payoff games and turned-based simple stochastic games =-=[11]-=-. Although now Zadeh’s well-known LEAST-ENTERED rule [27] and Cunningham’s ROUND-ROBIN rule are known to have subexponential lower bounds in concrete settings [14], there are still some historybased p...

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...lems is known to be in P, despite intensive research work on developing algorithms for them. For several of the existing algorithms, exponential lower bounds on their runtime have been given recently =-=[15, 16]-=-. Automatizability is an important concept for automated theorem proving. Call a propositional proof system automatizable if there is an algorithm which, given a tautology, produces a proof in time po...

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...stic improvement rule is the original locally improving rule that selects the best local improvement in every point [19]. However, we have shown that there is an exponential lower bound for this rule =-=[6, 7]-=-. The exponential lower bound is based on the implementation of a binary counter in the context of a parity game. Cyclic structures represent the bits of the counter with certain strategy settings cor...

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...mber of iterations for almost-sure ergodic games. Again our result is optimal, since even for turn-based stochastic reachability games the strategy-iteration algorithms require exponential iterations =-=[16, 14]-=-. We analyze the value-iteration algorithm for ergodic games and show that for all ǫ > 0, the value-iteration algorithm requires at most O(H ·W · ǫ−1 · log(ǫ−1)) iterations, where H is the upper bound...

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...e will use parity games where every colour is unique, i.e., where is injective. The reason for this assumption is that we extended [19], which implements variants of [31], and the lower bounds from =-=[13, 12, 14]-=- refer to such games. All parity games can be translated into such games as discussed in [31]. For these games, we use a valuation function based on their progress measure. We define ⌘ as hc0, c1, . ....

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...her they have exponential lower bounds. We consider the games described by Jurdziński that shows the exponential lower bound for small progress measures [42], the ladder games described by Friedmann =-=[27]-=- with the variation of recursive ladder games that give a lower bound for the strategy improvement algorithms [75, 63], and model checker ladder games [26] for which the algorithm by Stevens and Stirl...

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...́ski [122]. Later Schewe [104] has proposed a modification of the strategy improvement policy. Even for this improvement, Friedmann gives examples of games requiring exponential numbers of iterations =-=[52]-=-. It is actually not that surprising that the quest for polynomial time algorithm for model-checking is still on. The problem is closely related to other stubborn questions of a similar type, as for e...

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...G the sink existence property can be verified by standard graph algorithms. Given an initial strategy θ the sink seeking property can also be easily checked, as shown by the following lemma. Lemma 2 (=-=[8]-=-) Let G be a parity game with initial strategy θ fulfilling the sink existence property w.r.t. v∗. G is a sink game iff G is completely won by player 1 (i.e. W1 = V ) and for each node w it holds that...