Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
Tree automata, Mu-Calculus and determinacy
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Deciding the winner in parity games is in UP ∩ co-UP
Information Processing Letters
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
The Complexity of First-Order and Monadic Second-Order Logic Revisited
LICS '02 Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science
A Discrete Subexponential Algorithm for Parity Games
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Small Progress Measures for Solving Parity Games
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
A Discrete Strategy Improvement Algorithm for Solving Parity Games
CAV '00 Proceedings of the 12th International Conference on Computer Aided Verification
Automata logics, and infinite games: a guide to current research
Automata logics, and infinite games: a guide to current research
A deterministic subexponential algorithm for solving parity games
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
DAG-width: connectivity measure for directed graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
The Descriptive Complexity of Parity Games
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
The complexity of Nash equilibria in infinite multiplayer games
FOSSACS'08/ETAPS'08 Proceedings of the Theory and practice of software, 11th international conference on Foundations of software science and computational structures
The dag-width of directed graphs
Journal of Combinatorial Theory Series B
Parity games on undirected graphs
Information Processing Letters
Polynomial-Time algorithms for energy games with special weight structures
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Alternating traps in Muller and parity games
Theoretical Computer Science
Digraph width measures in parameterized algorithmics
Discrete Applied Mathematics
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The question of the exact complexity of solving parity games is one of the major open problems in system verification, as it is equivalent to the problem of model-checking the modal µ-calculus. The known upper bound is NP∩co-NP, but no polynomial algorithm is known. It was shown that on tree-like graphs (of bounded tree-width and DAG-width) a polynomial-time algorithm does exist. Here we present a polynomial-time algorithm for parity games on graphs of bounded clique-width (class of graphs containing e.g. complete bipartite graphs and cliques), thus completing the picture. This also extends the tree-width result, as graphs of bounded tree-width are a subclass of graphs of bounded clique-width. The algorithm works in a different way to the tree-width case and relies heavily on an interesting structural property of parity games.