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Approximating the Value of a Concurrent Reachability Game in the Polynomial Time Hierarchy

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Algorithms and Computation (ISAAC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8283))

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Abstract

We show that the value of a finite-state concurrent reachability game can be approximated to arbitrary precision in TFNP[NP], that is, in the polynomial time hierarchy. Previously, no better bound than PSPACE was known for this problem. The proof is based on formulating a variant of the state reduction algorithm for Markov chains using arbitrary precision floating point arithmetic and giving a rigorous error analysis of the algorithm.

The authors acknowledge support from The Danish National Research Foundation and The National Science Foundation of China (under the grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation and from the Center for research in the Foundations of Electronic Markets (CFEM), supported by the Danish Strategic Research Council.

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References

  1. Chatterjee, K., de Alfaro, L., Henzinger, T.A.: Strategy improvement for concurrent reachability games. In: Third International Conference on the Quantitative Evaluation of Systems, QEST 2006, pp. 291–300. IEEE Computer Society (2006)

    Google Scholar 

  2. Chatterjee, K., Majumdar, R., Jurdziński, M.: On Nash equilibria in stochastic games. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 26–40. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  3. de Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. Theor. Comput. Sci. 386(3), 188–217 (2007)

    Article  MATH  Google Scholar 

  4. Etessami, K., Yannakakis, M.: Recursive concurrent stochastic games. Logical Methods in Computer Science 4(4) (2008)

    Google Scholar 

  5. Everett, H.: Recursive games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games Vol. III, vol. 39, Annals of Mathematical Studies. Princeton University Press (1957)

    Google Scholar 

  6. Grassmann, W.K., Taksar, M.I., Heyman, D.P.: Regenerative analysis and steady state distributions for Markov chains. Operations Research, 1107–1116 (1985)

    Google Scholar 

  7. Hansen, K.A., Ibsen-Jensen, R., Miltersen, P.B.: The complexity of solving reachability games using value and strategy iteration. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 77–90. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  8. Hansen, K.A., Koucky, M., Miltersen, P.B.: Winning concurrent reachability games requires doubly exponential patience. In: 24th Annual IEEE Symposium on Logic in Computer Science (LICS 2009), pp. 332–341. IEEE (2009)

    Google Scholar 

  9. Himmelberg, C.J., Parthasarathy, T., Raghavan, T.E.S., Vleck, F.S.V.: Existence of p-equilibrium and optimal stationary strategies in stochastic games. Proc. Amer. Math. Soc. 60, 245–251 (1976)

    MathSciNet  Google Scholar 

  10. Megiddo, N., Papadimitriou, C.H.: On total functions, existence theorems and computational complexity. Theor. Comput. Sci. 81(2), 317–324 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. O’Cinneide, C.A.: Entrywise perturbation theory and error analysis for Markov chains. Numerische Mathematik 65(1), 109–120 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Parthasarathy, T.: Discounted and positive stochastic games. Bull. Amer. Math. Soc. 77, 134–136 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sheskin, T.J.: A Markov chain partitioning algorithm for computing steady state probabilities. Operations Research, 228–235 (1985)

    Google Scholar 

  14. Solan, E.: Continuity of the value of competitive Markov decision processes. Journal of Theoretical Probability 16, 831–845 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wilkinson, J.H.: A priori error analysis of algebraic processes. In: Proceedings International Congress Math., pp. 629–639. Izdat. Mir, Moscow (1968)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Aarhus University, Denmark

    Søren Kristoffer Stiil Frederiksen & Peter Bro Miltersen

Authors
  1. Søren Kristoffer Stiil Frederiksen
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  2. Peter Bro Miltersen
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, 100084, Hong Kong, China

    Leizhen Cai

  2. Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, 100044, Hong Kong, China

    Siu-Wing Cheng

  3. Department of Computer Science, University of Hong Kong, Porfkulam, 100084, Hong Kong, China

    Tak-Wah Lam

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Frederiksen, S.K.S., Miltersen, P.B. (2013). Approximating the Value of a Concurrent Reachability Game in the Polynomial Time Hierarchy. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_43

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  • DOI: https://doi.org/10.1007/978-3-642-45030-3_43

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45029-7

  • Online ISBN: 978-3-642-45030-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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