Parity Games of Bounded Tree-Depth

Author Konrad Staniszewski

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Konrad Staniszewski
  • University of Warsaw, Poland
  • IDEAS NCBR Sp. z o.o., Warsaw, Poland


I want to thank the supervisor of my master’s thesis Damian Niwiński and anonymous reviewers for valuable feedback.

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Konrad Staniszewski. Parity Games of Bounded Tree-Depth. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The exact complexity of solving parity games is a major open problem. Several authors have searched for efficient algorithms over specific classes of graphs. In particular, Obdržálek showed that for graphs of bounded tree-width or clique-width, the problem is in P, which was later improved by Ganardi, who showed that it is even in LOGCFL (with an additional assumption for clique-width case). Here we extend this line of research by showing that for graphs of bounded tree-depth the problem of solving parity games is in logspace uniform AC⁰. We achieve this by first considering a parameter that we obtain from a modification of clique-width, which we call shallow clique-width. We subsequently provide a suitable reduction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Parity Games
  • Circuits
  • Tree-Depth
  • Clique-Width


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  1. Miklós Ajtai. ∑^1_1-formulae on finite structures. Ann. Pure Appl. Log., 24(1):1-48, 1983. URL:
  2. Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. URL:
  3. Dietmar Berwanger, Anuj Dawar, Paul Hunter, and Stephan Kreutzer. Dag-width and parity games. In Bruno Durand and Wolfgang Thomas, editors, STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 23-25, 2006, Proceedings, volume 3884 of Lecture Notes in Computer Science, pages 524-536. Springer, 2006. URL:
  4. Dietmar Berwanger and Erich Grädel. Entanglement - A measure for the complexity of directed graphs with applications to logic and games. In Franz Baader and Andrei Voronkov, editors, Logic for Programming, Artificial Intelligence, and Reasoning, 11th International Conference, LPAR 2004, Montevideo, Uruguay, March 14-18, 2005, Proceedings, volume 3452 of Lecture Notes in Computer Science, pages 209-223. Springer, 2004. URL:
  5. Cristian S. Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and Frank Stephan. Deciding parity games in quasipolynomial time. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 252-263. ACM, 2017. URL:
  6. Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci., 46(2):218-270, 1993. URL:
  7. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discret. Appl. Math., 101(1-3):77-114, 2000. URL:
  8. Anuj Dawar and Erich Grädel. The descriptive complexity of parity games. In Michael Kaminski and Simone Martini, editors, Computer Science Logic, 22nd International Workshop, CSL 2008, 17th Annual Conference of the EACSL, Bertinoro, Italy, September 16-19, 2008. Proceedings, volume 5213 of Lecture Notes in Computer Science, pages 354-368. Springer, 2008. URL:
  9. Reinhard Diestel. Graph Theory. Springer Publishing Company, Incorporated, 5th edition, 2017. URL:
  10. Michael Elberfeld, Martin Grohe, and Till Tantau. Where first-order and monadic second-order logic coincide. ACM Trans. Comput. Log., 17(4):25, 2016. URL:
  11. Michael Elberfeld, Andreas Jakoby, and Till Tantau. Algorithmic meta theorems for circuit classes of constant and logarithmic depth. In Christoph Dürr and Thomas Wilke, editors, 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, volume 14 of LIPIcs, pages 66-77. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL:
  12. E. Allen Emerson and Charanjit S. Jutla. Tree automata, mu-calculus and determinacy (extended abstract). In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1-4 October 1991, pages 368-377. IEEE Computer Society, 1991. URL:
  13. John Fearnley and Sven Schewe. Time and parallelizability results for parity games with bounded treewidth. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part II, volume 7392 of Lecture Notes in Computer Science, pages 189-200. Springer, 2012. URL:
  14. Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory, 17(1):13-27, 1984. URL:
  15. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Gregory Z. Gutin and Stefan Szeider, editors, Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, volume 8246 of Lecture Notes in Computer Science, pages 163-176. Springer, 2013. URL:
  16. Moses Ganardi. Parity games of bounded tree- and clique-width. In Andrew M. Pitts, editor, Foundations of Software Science and Computation Structures - 18th International Conference, FoSSaCS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, London, UK, April 11-18, 2015. Proceedings, volume 9034 of Lecture Notes in Computer Science, pages 390-404. Springer, 2015. URL:
  17. Robert Ganian, Petr Hlinený, Joachim Kneis, Alexander Langer, Jan Obdrzálek, and Peter Rossmanith. Digraph width measures in parameterized algorithmics. Discret. Appl. Math., 168:88-107, 2014. URL:
  18. Robert Ganian, Petr Hlinený, Jaroslav Nesetril, Jan Obdrzálek, and Patrice Ossona de Mendez. Shrub-depth: Capturing height of dense graphs. Log. Methods Comput. Sci., 15(1), 2019. URL:
  19. Robert Ganian, Petr Hlinený, Jaroslav Nesetril, Jan Obdrzálek, Patrice Ossona de Mendez, and Reshma Ramadurai. When trees grow low: Shrubs and fast MSO1. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Bratislava, Slovakia, August 27-31, 2012. Proceedings, volume 7464 of Lecture Notes in Computer Science, pages 419-430. Springer, 2012. URL:
  20. Paul Hunter and Stephan Kreutzer. Digraph measures: Kelly decompositions, games, and orderings. Theor. Comput. Sci., 399(3):206-219, 2008. URL:
  21. Marcin Jurdzinski and Ranko Lazic. Succinct progress measures for solving parity games. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017, pages 1-9. IEEE Computer Society, 2017. URL:
  22. Karoliina Lehtinen, Pawel Parys, Sven Schewe, and Dominik Wojtczak. A recursive approach to solving parity games in quasipolynomial time. Log. Methods Comput. Sci., 18(1), 2022. URL:
  23. Michael Luttenberger, Philipp J. Meyer, and Salomon Sickert. Practical synthesis of reactive systems from ltl specifications via parity games. Acta Informatica, 57(1–2):3-36, November 2019. URL:
  24. Jaroslav Nesetril and Patrice Ossona de Mendez. Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb., 27(6):1022-1041, 2006. URL:
  25. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL:
  26. Jan Obdrzálek. Fast mu-calculus model checking when tree-width is bounded. In Warren A. Hunt Jr. and Fabio Somenzi, editors, Computer Aided Verification, 15th International Conference, CAV 2003, Boulder, CO, USA, July 8-12, 2003, Proceedings, volume 2725 of Lecture Notes in Computer Science, pages 80-92. Springer, 2003. URL:
  27. Jan Obdrzálek. Clique-width and parity games. In Jacques Duparc and Thomas A. Henzinger, editors, Computer Science Logic, 21st International Workshop, CSL 2007, 16th Annual Conference of the EACSL, Lausanne, Switzerland, September 11-15, 2007, Proceedings, volume 4646 of Lecture Notes in Computer Science, pages 54-68. Springer, 2007. URL:
  28. Konrad Staniszewski. Parity games of bounded tree-depth, 2022. URL: