Skip to main content

Time and Parallelizability Results for Parity Games with Bounded Treewidth

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7392)

Abstract

Parity games are a much researched class of games in NP ∩ CoNP that are not known to be in P. Consequently, researchers have considered specialised algorithms for the case where certain graph parameters are small. In this paper, we show that, if a tree decomposition is provided, then parity games with bounded treewidth can be solved in O(k 3k + 2 ·n 2 ·(d + 1)3k) time, where nk, and d are the size, treewidth, and number of priorities in the parity game. This significantly improves over previously best algorithm, given by Obdržálek, which runs in \(O(n \cdot d^{2(k+1)^2})\) time. Our techniques can also be adapted to show that the problem lies in the complexity class NC2, which is the class of problems that can be efficiently parallelized. This is in stark contrast to the general parity game problem, which is known to be P-hard, and thus unlikely to be contained in NC.

Keywords

  • Model Check
  • Turing Machine
  • Input Graph
  • Tree Decomposition
  • Winning Strategy

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allender, E., Loui, M.C., Regan, K.W.: Complexity classes. In: Atallah, M.J., Blanton, M. (eds.) Algorithms and Theory of Computation Handbook, p. 22. Chapman & Hall/CRC (2010)

    Google Scholar 

  2. Amir, E.: Efficient approximation for triangulation of minimum treewidth. In: Proc. of UAI, pp. 7–15. Morgan Kaufmann Publishers Inc., San Francisco (2001)

    Google Scholar 

  3. 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)

    CrossRef  Google Scholar 

  4. 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)

    CrossRef  Google Scholar 

  5. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25(6), 1305–1317 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Emerson, E.A., Jutla, C.S.: Tree automata, μ-calculus and determinacy. In: Proc. of FOCS, pp. 368–377. IEEE Computer Society Press (October 1991)

    Google Scholar 

  7. Emerson, E.A., Jutla, C.S., Sistla, A.P.: On Model-Checking for Fragments of μ-Calculus. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 385–396. Springer, Heidelberg (1993)

    CrossRef  Google Scholar 

  8. 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)

    CrossRef  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Mostowski, A.W.: Games with forbidden positions. Technical Report 78, University of Gdańsk (1991)

    Google Scholar 

  11. 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)

    CrossRef  Google Scholar 

  12. Stirling, C.: Local Model Checking Games (Extended Abstract). In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)

    CrossRef  Google Scholar 

  13. Thorup, M.: All structured programs have small tree width and good register allocation. Information and Computation 142(2), 159–181 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Fearnley, J., Schewe, S. (2012). Time and Parallelizability Results for Parity Games with Bounded Treewidth. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31585-5_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31585-5_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31584-8

  • Online ISBN: 978-3-642-31585-5

  • eBook Packages: Computer ScienceComputer Science (R0)