Abstract
Determining the winner of a Parity Game is a major problem in computational complexity with a number of applications in verification. In a parameterized complexity setting, the problem has often been considered with parameters such as (directed versions of) treewidth or clique-width, by applying dynamic programming techniques.
In this paper we adopt a parameterized approach which is more inspired by well-known (non-parameterized) algorithms for this problem. We consider a number of natural parameterizations, such as by Directed Feedback Vertex Set, Distance to Tournament, and Modular Width. We show that, for these parameters, it is possible to obtain recursive parameterized algorithms which are simpler, faster and only require polynomial space. We complement these results with some algorithmic lower bounds which, among others, rule out a possible avenue for improving the best-known sub-exponential time algorithm for parity games.
J. Gajarský—Supported by the research centre Institute for Theoretical Computer Science (CE-ITI), project P202/12/G061.
V. Mitsou—Supported by ERC Starting Grant PARAMTIGHT (No. 280152).
S. Ordyniak—Supported by Employment of Newly Graduated Doctors of Science for Scientific Excellence (CZ.1.07/2.3.00/30.0009).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As usual in parameterized complexity the \(O^*\!\!\left( f(k)\right) \)-notation, for some function f of the parameter k, means that there is an algorithm running in time \(O(f(k)n^{O(1)})\), where n is the input size of the problem.
References
Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S., Obdržálek, J.: The dag-width of directed graphs. J. Comb. Theo. Ser. B 102(4), 900–923 (2012)
Berwanger, D., Grädel, E., Kaiser, L., Rabinovich, R.: Entanglement and the complexity of directed graphs. Theor. Comput. Sci. 463, 2–25 (2012)
Bjorklund, H., Sandberg, S., Vorobyov, S.: On fixed-parameter complexity of infinite games. In: Sere, K., Waldén, M. (eds.) The Nordic Workshop on Programming Theory (NWPT 2003), number 34 in Åbo Akademi, Reports on Computer Science and Mathematics, pp. 29–31. Citeseer (2003)
Björklund, H., Sandberg, S., Vorobyov, S.G.: Memoryless determinacy of parity and mean payoff games: a simple proof. Theor. Comput. Sci. 310(1–3), 365–378 (2004)
Björklund, H., Vorobyov, S.G.: A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games. Discrete Appl. Math. 155(2), 210–229 (2007)
Bojanczyk, M., Dittmann, C., Kreutzer, S.: Decomposition theorems and model-checking for the modal \(\mu \)-calculus. In: Henzinger, T.A., Miller, D. (eds.) Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS 2014, Vienna, Austria, July 14–18, 2014, p. 17. ACM (2014)
Browne, A., Clarke, E.M., Jha, S., Long, D.E., Marrero, W.R.: An improved algorithm for the evaluation of fixpoint expressions. Theor. Comput. Sci. 178(1–2), 237–255 (1997)
Chatterjee, K., Henzinger, T.A.: A survey of stochastic \(\omega \)-regular games. J. Comput. Syst. Sci. 78(2), 394–413 (2012)
Condon, A.: The complexity of stochastic games. Inf. Comput. 96(2), 203–224 (1992)
Dittmann, C., Kreutzer, S., Tomescu, A.I.: Graph operations on parity games and polynomial-time algorithms. arXiv preprint (2012). arXiv:1208.1640
Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM J. Comput. 29(1), 132–158 (1999)
Emerson, E.A., Jutla, C.S., Sistla, A.P.: On model checking for the \(\mu \)-calculus and its fragments. Theor. Comput. Sci. 258(1–2), 491–522 (2001)
Fomin, F.V., Liedloff, M., Montealegre, P., Todinca, I.: Algorithms parameterized by vertex cover and modular width, through potential maximal cliques. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 182–193. Springer, Heidelberg (2014)
Gajarský, J., Lampis, M., Ordyniak, S.: Parameterized algorithms for modular-width. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 163–176. Springer, Heidelberg (2013)
Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research. LNCS, vol. 2500. Springer, Heidelberg (2002)
Hunter, P., Kreutzer, S.: Digraph measures: kelly decompositions, games, and orderings. Theor. Comput. Sci. 399(3), 206–219 (2008)
Jurdzinski, M.: Deciding the winner in parity games is in UP cap co-up. Inf. Process. Lett. 68(3), 119–124 (1998)
Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000)
Jurdzinski, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38(4), 1519–1532 (2008)
Küsters, R.: Memoryless determinacy of parity games. In: Grädel, E., Thomas, W., Wilke, T. (eds.) Automata Logics, and Infinite Games. LNCS, vol. 2500, pp. 95–106. Springer, Heidelberg (2002)
McNaughton, R.: Infinite games played on finite graphs. Ann. Pure Appl. Logic 65(2), 149–184 (1993)
Obdržálek, J.: Fast mu-calculus model checking when tree-width is bounded. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 80–92. Springer, Heidelberg (2003)
Obdržálek, J.: Clique-width and parity games. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 54–68. Springer, Heidelberg (2007)
Schewe, S.: Solving parity games in big steps. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 449–460. Springer, Heidelberg (2007)
Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200(1–2), 135–183 (1998)
Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1&2), 343–359 (1996)
Acknowledgements
We would like to thank Danupon Nanongkai for suggesting this problem and for our useful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gajarský, J., Lampis, M., Makino, K., Mitsou, V., Ordyniak, S. (2015). Parameterized Algorithms for Parity Games. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_28
Download citation
DOI: https://doi.org/10.1007/978-3-662-48054-0_28
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48053-3
Online ISBN: 978-3-662-48054-0
eBook Packages: Computer ScienceComputer Science (R0)