The Theory of Universal Graphs for Infinite Duration Games

Abstract

We introduce the notion of universal graphs as a tool for constructing algorithms solving games of infinite duration such as parity games and mean payoff games. In the first part we develop the theory of universal graphs, with two goals: showing an equivalence and normalisation result between different recently introduced related models, and constructing generic value iteration algorithms for any positionally determined objective. In the second part we give four applications: to parity games, to mean payoff games, to a disjunction between a parity and a mean payoff objective, and to disjunctions of several mean payoff objectives. For each of these four cases we construct algorithms achieving or improving over the best known time and space complexity.

Playing Safe, Ten Years Later

“…A promising line of work extending the approach initiated in this paper uses the notion of universal graphs, initially introduced for understanding algorithms solving games [CFGO22], and later used by Ohlmann for giving a characterisation of all positionally determined The subsequent works most related to the present results are by Bouyer, Casares, Randour and Vandenhove [BCRV22], and by Bouyer, Fijalkow, Randour, and Vandenhove [BFRV23]. The first give a characterization of half-positional objectives recognized by deterministic Büchi automata, and the second characterizations of chromatic memory requirements for open and closed objectives, as well as complexity-theoretic results about computing these requirements.…”

Rabin Games and Colourful Universal Trees

“…Inspired by the breakthrough algorithms to solve parity games, Colcombet, Fijalkow, Gawrychowski, and Ohlmann [5] proposed a formalism for algorithms that solve games where one player has a positional strategy. They showed that if there is a special kind of graph homomorphism into a graph with a total order on its vertices, then one can obtain a lifting algorithm for such games.…”