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Clique-Width and Parity Games

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Book cover Computer Science Logic (CSL 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4646))

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Abstract

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.

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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.: Entanglement – a measure for the complexity of directed graphs with applications to logic and games. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 209–223. Springer, Heidelberg (2005)

    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)

    Google Scholar 

  4. Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science: Volume B: Formal Models and Semantics, pp. 193–242. Elsevier, Amsterdam (1990)

    Google Scholar 

  5. Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1-3), 77–114 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy. In: Proc. 5th IEEE Foundations of Computer Science, pp. 368–377. IEEE Computer Society Press, Los Alamitos (1991)

    Google Scholar 

  8. Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. In: LICS 2002, pp. 215–224. IEEE Computer Society Press, Los Alamitos (2002)

    Google Scholar 

  9. Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  10. Jurdziński, M.: Deciding the winner in parity games is in UP ∩ co-UP. Information Processing Letters 68(3), 119–124 (1998)

    Article  MathSciNet  Google Scholar 

  11. 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 

  12. Jurdziński, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms, pp. 117–123. ACM Press, New York (2006)

    Chapter  Google Scholar 

  13. Kozen, D.: Results on the propositional μ-calculus. Theoretical Computer Science 27, 333–354 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mostowski, A.W.: Games with forbidden positions. Technical Report 78, Instytut Matematyki, Uniwersytet Gdański, Poland (1991)

    Google Scholar 

  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)

    Google Scholar 

  16. Obdržálek, J.: Algorithmic Analysis of Parity Games. PhD thesis, University of Edinburgh (2006)

    Google Scholar 

  17. Obdržálek, J.: DAG-width – connectivity measure for directed graphs. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms, pp. 814–821. ACM Press, New York (2006)

    Chapter  Google Scholar 

  18. Robertson, N., Seymour, P.D.: Graph Minors. III. Planar tree-width. Journal of Combinatorial Theory, Series B 36, 49–63 (1984)

    Article  MATH  MathSciNet  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 

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Authors and Affiliations

  1. Faculty of Informatics, Masaryk University, Brno, Czech Republic

    Jan Obdržálek

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  1. Jan Obdržálek
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Jacques Duparc Thomas A. Henzinger

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Obdržálek, J. (2007). Clique-Width and Parity Games. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_8

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  • DOI: https://doi.org/10.1007/978-3-540-74915-8_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74914-1

  • Online ISBN: 978-3-540-74915-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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