Parameterized complexity of games with monotonically ordered omega-regular objectives

Authors Véronique Bruyère, Quentin Hautem, Jean-François Raskin

Thumbnail PDF


  • Filesize: 0.53 MB
  • 16 pages

Document Identifiers

Author Details

Véronique Bruyère
  • Département d'informatique, Université de Mons (UMONS), Mons, Belgium
Quentin Hautem
  • Département d'informatique, Université de Mons (UMONS), Mons, Belgium
Jean-François Raskin
  • Département d'informatique, Université Libre de Bruxelles (ULB), Brussels, Belgium

Cite AsGet BibTex

Véronique Bruyère, Quentin Hautem, and Jean-François Raskin. Parameterized complexity of games with monotonically ordered omega-regular objectives. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


In recent years, two-player zero-sum games with multiple objectives have received a lot of interest as a model for the synthesis of complex reactive systems. In this framework, Player 1 wins if he can ensure that all objectives are satisfied against any behavior of Player 2. When this is not possible to satisfy all the objectives at once, an alternative is to use some preorder on the objectives according to which subset of objectives Player 1 wants to satisfy. For example, it is often natural to provide more significance to one objective over another, a situation that can be modelled with lexicographically ordered objectives for instance. Inspired by recent work on concurrent games with multiple omega-regular objectives by Bouyer et al., we investigate in detail turned-based games with monotonically ordered and omega-regular objectives. We study the threshold problem which asks whether player 1 can ensure a payoff greater than or equal to a given threshold w.r.t. a given monotonic preorder. As the number of objectives is usually much smaller than the size of the game graph, we provide a parametric complexity analysis and we show that our threshold problem is in FPT for all monotonic preorders and all classical types of omega-regular objectives. We also provide polynomial time algorithms for Büchi, coBüchi and explicit Muller objectives for a large subclass of monotonic preorders that includes among others the lexicographic preorder. In the particular case of lexicographic preorder, we also study the complexity of computing the values and the memory requirements of optimal strategies.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Algorithmic game theory
  • two-player zero-sum games played on graphs
  • omega-regular objectives
  • ordered objectives
  • parameterized complexity


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Shaull Almagor and Orna Kupferman. Latticed-LTL synthesis in the presence of noisy inputs. Discrete Event Dynamic Systems, 27(3):547-572, 2017. URL:
  2. Rajeev Alur, Aditya Kanade, and Gera Weiss. Ranking automata and games for prioritized requirements. In CAV Proceedings, volume 5123 of LNCS, pages 240-253. Springer, 2008. URL:
  3. Rajeev Alur, Salvatore La Torre, and P. Madhusudan. Playing games with boxes and diamonds. In CONCUR Proceedings, volume 2761 of LNCS, pages 127-141. Springer, 2003. URL:
  4. Roderick Bloem, Krishnendu Chatterjee, Karin Greimel, Thomas A. Henzinger, and Barbara Jobstmann. Robustness in the presence of liveness. In CAV Proceedings, volume 6174 of LNCS, pages 410-424. Springer, 2010. URL:
  5. Roderick Bloem, Krishnendu Chatterjee, Thomas A. Henzinger, and Barbara Jobstmann. Better quality in synthesis through quantitative objectives. In CAV Proceedings, volume 5643 of LNCS, pages 140-156. Springer, 2009. URL:
  6. Patricia Bouyer, Romain Brenguier, Nicolas Markey, and Michael Ummels. Concurrent games with ordered objectives. In FOSSACS Proceedings, volume 7213 of LNCS, pages 301-315. Springer, 2012. URL:
  7. Véronique Bruyère, Quentin Hautem, and Jean-François Raskin. On the complexity of heterogeneous multidimensional games. In CONCUR Proceedings, volume 59 of LIPIcs, pages 11:1-11:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL:
  8. Véronique Bruyère, Noémie Meunier, and Jean-François Raskin. Secure equilibria in weighted games. In CSL-LICS Proceedings, pages 26:1-26:26. ACM, 2014. URL:
  9. Cristian S. Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and Frank Stephan. Deciding parity games in quasipolynomial time. In STOC Proceedings, pages 252-263. ACM, 2017. URL:
  10. Krishnendu Chatterjee, Wolfgang Dvorák, Monika Henzinger, and Veronika Loitzenbauer. Conditionally optimal algorithms for generalized Büchi games. In MFCS Proceedings, volume 58 of LIPIcs, pages 25:1-25:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL:
  11. Krishnendu Chatterjee, Thomas A. Henzinger, and Nir Piterman. Generalized parity games. In FOSSACS Proceedings, volume 4423 of LNCS, pages 153-167. Springer, 2007. URL:
  12. Thomas Colcombet, Marcin Jurdzinski, Ranko Lazic, and Sylvain Schmitz. Perfect half space games. In LICS Proceedings, pages 1-11. IEEE Computer Society, 2017. URL:
  13. Rodney G. Downey and M. R. Fellows. Parameterized Complexity. Springer Publishing Company, Incorporated, 2012. Google Scholar
  14. Nathanaël Fijalkow and Florian Horn. Les jeux d'accessibilité généralisée. Technique et Science Informatiques, 32:931-949, 2013. URL:
  15. Erich Grädel, Wolfgang Thomas, and Thomas Wilke, editors. Automata, Logics, and Infinite Games: A Guide to Current Research [outcome of a Dagstuhl seminar, February 2001], volume 2500 of LNCS. Springer, 2002. Google Scholar
  16. Julian Gutierrez, Aniello Murano, Giuseppe Perelli, Sasha Rubin, and Michael Wooldridge. Nash equilibria in concurrent games with lexicographic preferences. In IJCAI Proceedings, pages 1067-1073., 2017. URL:
  17. Florian Horn. Explicit Muller games are PTIME. In FSTTCS Proceedings, volume 2 of LIPIcs, pages 235-243. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2008. URL:
  18. Paul Hunter and Anuj Dawar. Complexity bounds for regular games. In MFCS Proceedings, volume 3618 of LNCS, pages 495-506. Springer, 2005. URL:
  19. Orna Kupferman, Giuseppe Perelli, and Moshe Y. Vardi. Synthesis with rational environments. In EUMAS Proceedings, volume 8953 of LNCS, pages 219-235. Springer, 2014. URL:
  20. Donald A. Martin. Borel determinacy. Annals of Mathematics, 102:363-371, 1975. Google Scholar
  21. A. Pnueli and R. Rosner. On the synthesis of a reactive module. In POPL Proceedings, pages 179-190. ACM Press, 1989. Google Scholar
  22. Yaron Velner, Krishnendu Chatterjee, Laurent Doyen, Thomas A. Henzinger, Alexander Moshe Rabinovich, and Jean-François Raskin. The complexity of multi-mean-payoff and multi-energy games. Inf. Comput., 241:177-196, 2015. URL:
  23. Uri Zwick and Mike Paterson. The complexity of mean payoff games on graphs. Theor. Comput. Sci., 158, 959:343-359, 1996. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail