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New Deterministic Algorithms for Solving Parity Games

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LATIN 2016: Theoretical Informatics (LATIN 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9644))

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Abstract

We study parity games in which one of the two players controls only a small number k of nodes and the other player controls the \(n-k\) other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time \(k^{O(\sqrt{k})}\cdot O(n^3)\) and general parity games in time \((p+k)^{O(\sqrt{k})} \cdot O(pnm)\), where p denotes the number of distinct priorities and m denotes the number of edges. For all games with \(k = o(n)\) this improves the previously fastest algorithm by Jurdziński, Paterson, and Zwick (SICOMP 2008).

We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree.

This research was supported by ERC Starting Grant 306465 (BeyondWorstCase).

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References

  1. Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: DAG-width and parity games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  2. Berwanger, D., Grädel, E., Kaiser, Ł., Rabinovich, R.: Entanglement and the complexity of directed graphs. Theoret. Comput. Sci. 463, 2–25 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Björklund, H., Sandberg, S., Vorobyov, S.: A discrete subexponential algorithm for parity games. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 663–674. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  4. Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy. In: Proceedings of FOCS 1991, pp. 368–377 (1991)

    Google Scholar 

  5. Fearnley, J., Lachish, O.: Parity games on graphs with medium tree-width. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 303–314. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  6. Fearnley, J., Schewe, S.: Time and parallelizability results for parity games with bounded treewidth. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 189–200. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  7. Gajarský, J., Lampis, M., Makino, K., Mitsou, V., Ordyniak, S.: Parameterized algorithms for parity games. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9235, pp. 336–347. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  8. Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research. LNCS, vol. 2500. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  9. Jurdziński, M.: Deciding the winner in parity games is in \(\rm UP\cap co\)-\(\rm UP\). Inform. Process. Lett. 68(3), 119–124 (1998)

    Article  MathSciNet  Google Scholar 

  10. 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)

    Chapter  Google Scholar 

  11. Jurdziński, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38(4), 1519–1532 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kloks, T., Bodlaender, H.L.: On the treewidth and pathwidth of permutation graphs (1992). http://www.cs.uu.nl/research/techreps/repo/CS-1992/1992-13.pdf

  13. McNaughton, R.: Infinite games played on finite graphs. Ann. Pure Appl. Logic 65(2), 149–184 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mnich, M., Röglin, H., Rösner, C.: New deterministic algorithms for solving parity games. arXiv.org, cs.CC, December 2015. http://arxiv.org/abs/1512.03246

  15. 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)

    Chapter  Google Scholar 

  16. 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)

    Chapter  Google Scholar 

  17. 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)

    Chapter  Google Scholar 

  18. Stirling, C.: Local model checking games. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)

    Google Scholar 

  19. Vöge, J., Jurdziński, M.: A discrete strategy improvement algorithm for solving parity games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  20. Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoret. Comput. Sci. 200(1–2), 135–183 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

M.M. thanks Lászlo Végh for introducing him to parity games, and the authors of [7] for sending us a preprint.

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Authors and Affiliations

  1. Department of Computer Science, University of Bonn, Bonn, Germany

    Matthias Mnich, Heiko Röglin & Clemens Rösner

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  1. Matthias Mnich
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  2. Heiko Röglin
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  3. Clemens Rösner
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Corresponding author

Correspondence to Clemens Rösner .

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Editors and Affiliations

  1. Carleton University, Ottawa, ON, Ontario, Canada

    Evangelos Kranakis

  2. University Chile, Santiago, Chile

    Gonzalo Navarro

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Mnich, M., Röglin, H., Rösner, C. (2016). New Deterministic Algorithms for Solving Parity Games. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_47

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  • DOI: https://doi.org/10.1007/978-3-662-49529-2_47

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