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Conference papers

Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games

Xavier Allamigeon 1 Stéphane Gaubert 1 Ricardo David Katz Mateusz Skomra 2 
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
Abstract : We develop value iteration-based algorithms to solve in a unified manner different classes of combinatorial zero-sum games with mean-payoff type rewards. These algorithms rely on an oracle, evaluating the dynamic programming operator up to a given precision. We show that the number of calls to the oracle needed to determine exact optimal (positional) strategies is, up to a factor polynomial in the dimension, of order R/sep, where the "separation" sep is defined as the minimal difference between distinct values arising from strategies, and R is a metric estimate, involving the norm of approximate sub and super-eigenvectors of the dynamic programming operator. We illustrate this method by two applications. The first one is a new proof, leading to improved complexity estimates, of a theorem of Boros, Elbassioni, Gurvich and Makino, showing that turn-based mean-payoff games with a fixed number of random positions can be solved in pseudo-polynomial time. The second one concerns entropy games, a model introduced by Asarin, Cervelle, Degorre, Dima, Horn and Kozyakin. The rank of an entropy game is defined as the maximal rank among all the ambiguity matrices determined by strategies of the two players. We show that entropy games with a fixed rank, in their original formulation, can be solved in polynomial time, and that an extension of entropy games incorporating weights can be solved in pseudo-polynomial time under the same fixed rank condition.
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Contributor : Mateusz Skomra Connect in order to contact the contributor
Submitted on : Tuesday, June 21, 2022 - 6:08:32 PM
Last modification on : Monday, July 4, 2022 - 10:00:51 AM


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  • HAL Id : hal-03698207, version 1


Xavier Allamigeon, Stéphane Gaubert, Ricardo David Katz, Mateusz Skomra. Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games. 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), Jul 2022, Paris, France. ⟨hal-03698207⟩



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