Cyclic games and an algorithm to find minimax cycle means in directed graphs

Gurvich, Vladimir; Karzanov, Alexander V.; Khachivan, L. G.

doi:10.1016/0041-5553(88)90012-2

Cited by 129 publications

(162 citation statements)

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“…It is not clear how the algorithm given in [GKK88] for the BW-case can be generalized to BWR-case. The proof in [BEGM09a] follows by considering the discounted case and then taking the discount factor β to the limit.…”

Section: Potential Transformations and Canonical Forms; Sketch Of Ourmentioning

confidence: 99%

“…This class of the BWR-games was introduced in [GKK88]; see also [CH08]. It was recently shown in [BEGM09a]) that the BWR-games and classical Gillette games [Gil57] are polynomially equivalent.…”

Section: Bwr-gamesmentioning

confidence: 99%

“…They were introduced for the complete bipartite digraphs in [Mou76b,Mou76a], for all (not necessarily complete) bipartite digraphs in [EM79], and for arbitrary digraphs in [GKK88]. A more special case was considered extensively in the literature under the name of parity games [BV01a,BV01b,CJH04,Hal07,Jur98,JPZ06], and later generalized also to include random nodes in [CH08].…”

Section: Bwr-gamesmentioning

confidence: 99%

“…It is known that for BW-games there exists a potential transformation such that, in the obtained game the locally optimal strategies are globally optimal, and hence, the value and optimal strategies become obvious [GKK88]. This result was extended for the more general class of BWR-games in [BEGM09a]: in the transformed game, the equilibrium value µ(v) = µ x (v) is given simply by the maximum local reward for v ∈ V W , the minimum local reward for v ∈ V B , and the average local reward for v ∈ V R .…”

Section: Potential Transformations and Canonical Forms; Sketch Of Ourmentioning

confidence: 99%

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A Pumping Algorithm for Ergodic Stochastic Mean Payoff Games with Perfect Information

Boros

Elbassioni

Gurvich

et al. 2010

Integer Programming and Combinatorial Optimization

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Abstract. We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph G = (V = VB ∪ VW ∪ VR, E), with local rewards r : E → R, and three types of vertices: black VB, white VW , and random VR. The game is played by two players, White and Black: When the play is at a white (black) vertex v, White (Black) selects an outgoing arc (v, u). When the play is at a random vertex v, a vertex u is picked with the given probability p (v, u). In all cases, Black pays White the value r(v, u). The play continues forever, and White aims to maximize (Black aims to minimize) the limiting mean (that is, average) payoff. It was recently shown in [BEGM09a] that BWR-games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games (SSG), stochastic parity games, and Markov decision processes. In this paper, we give a new algorithm for solving BWR-games in the ergodic case, that is when the optimal values do not depend on the initial position. Our algorithm solves a BWR-game by reducing it, using a potential transformation, to a canonical form in which the value and optimal strategies of both players are obvious for every initial position, since a locally optimal move in it is optimal in the whole game. We show that this algorithm is pseudo-polynomial when the number of random nodes is constant. We also provide an almost matching lower bound on its running time, and show that this bound holds for a wider class of algorithms. Our preliminary experiments with the algorithm indicate that it behaves much better than its estimated worst-case running time. This and the fact that every iteration is simple and can be implemented in a distributed way makes it a good candidate for practical applications.

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Section: Potential Transformations and Canonical Forms; Sketch Of Ourmentioning

confidence: 99%

Section: Bwr-gamesmentioning

confidence: 99%

Section: Bwr-gamesmentioning

confidence: 99%

Section: Potential Transformations and Canonical Forms; Sketch Of Ourmentioning

confidence: 99%

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A Pumping Algorithm for Ergodic Stochastic Mean Payoff Games with Perfect Information

Boros

Elbassioni

Gurvich

et al. 2010

Integer Programming and Combinatorial Optimization

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“…A natural restriction of stochastic mean-payoff games is to deterministic transitions (i.e., all probability distributions put all probability mass on one position). This class of games has been studied in the computer science literature under the names of cyclic games [11] and mean-payoff games [19]. We shall refer to them as deterministic mean-payoff games in this paper.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Solving Stochastic Games on Graphs

Andersson

Miltersen

2009

Lecture Notes in Computer Science

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Abstract. We consider some well-known families of two-player zero-sum perfect-information stochastic games played on finite directed graphs. Generalizing and unifying results of Liggett and Lippman, Zwick and Paterson, and Chatterjee and Henzinger, we show that the following tasks are polynomial-time (Turing) equivalent.-Solving stochastic parity games, -Solving simple stochastic games, -Solving stochastic terminal-payoff games with payoffs and probabilities given in unary, -Solving stochastic terminal-payoff games with payoffs and probabilities given in binary, -Solving stochastic mean-payoff games with rewards and probabilities given in unary, -Solving stochastic mean-payoff games with rewards and probabilities given in binary, -Solving stochastic discounted-payoff games with discount factor, rewards and probabilities given in binary. It is unknown whether these tasks can be performed in polynomial time. In the above list, "solving" may mean either quantitatively solving a game (computing the values of its positions) or strategically solving a game (computing an optimal strategy for each player). In particular, these two tasks are polynomial-time equivalent for all the games listed above. We also consider a more refined notion of equivalence between quantitatively and strategically solving a game. We exhibit a linear time algorithm that given a simple stochastic game or a terminal-payoff game and the values of all positions of that game, computes a pair of optimal strategies. Consequently, for any restriction one may put on the simple stochastic game model, quantitatively solving is polynomial-time equivalent to strategically solving the resulting class of games.

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