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Improved Pseudo-polynomial Bound for the Value Problem and Optimal Strategy Synthesis in Mean Payoff Games

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Abstract

In this work we offer an \(O(|V|^2 |E|\, W)\) pseudo-polynomial time deterministic algorithm for solving the Value Problem and Optimal Strategy Synthesis in Mean Payoff Games. This improves by a factor \(\log (|V|\, W)\) the best previously known pseudo-polynomial time upper bound due to Brim et al. The improvement hinges on a suitable characterization of values, and a description of optimal positional strategies, in terms of reweighted Energy Games and Small Energy-Progress Measures.

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References

  1. Andersson, D., Vorobyov, S.: Fast algorithms for monotonic discounted linear programs with two variables per inequality. Tech. rep., Preprint NI06019-LAA, Isaac Netwon Institute for Mathematical Sciences, Cambridge, UK (2006)

  2. Bouyer, P., Fahrenberg, U., Larsen, K., Markey, N., Srba, J.: Infinite runs in weighted timed automata with energy constraints. In: Cassez, F., Jard, C. (eds.) Formal Modeling and Analysis of Timed Systems, Lecture Notes in Computer Science, vol. 5215, pp. 33–47. Springer, Berlin (2008)

    Chapter  Google Scholar 

  3. Brim, L., Chaloupka, J., Doyen, L., Gentilini, R., Raskin, J.: Faster algorithms for mean-payoff games. Form. Methods Syst. Des. 38(2), 97–118 (2011)

    Article  MATH  Google Scholar 

  4. Chakrabarti, A., de Alfaro, L., Henzinger, T., Stoelinga, M.: Resource interfaces. In: Alur, R., Lee, I. (eds.) Embedded Software, Lecture Notes in Computer Science, vol. 2855, pp. 117–133. Springer, Berlin (2003)

    Google Scholar 

  5. Ehrenfeucht, A., Mycielski, J.: Positional strategies for mean payoff games. Int. J. Game Theory 8(2), 109–113 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley Longman Publishing Co., Inc, Boston (1994)

    MATH  Google Scholar 

  7. Gurvich, V., Karzanov, A., Khachiyan, L.: Cyclic games and an algorithm to find minimax cycle means in directed graphs. USSR Comput. Math. Math. Phys. 28(5), 85–91 (1988)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Lifshits, Y., Pavlov, D.: Potential theory for mean payoff games. J. Math. Sci. 145(3), 4967–4974 (2007)

    Article  MathSciNet  Google Scholar 

  10. Pawlewicz, J., Pătraşcu, M.: Order statistics in the farey sequences in sublinear time and counting primitive lattice points in polygons. Algorithmica 55(2), 271–282 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158, 343–359 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported by Department of Computer Science, University of Verona, Verona, Italy, under Ph.D. Grant “Computational Mathematics and Biology”, on a co-tutelle agreement with LIGM, Université Paris-Est in Marne-la-Vallée, Paris, France.

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Authors and Affiliations

  1. Department of Mathematics, University of Trento, Trento, Italy

    Carlo Comin

  2. LIGM, Université Paris-Est, Marne-la-Vallée, Paris, France

    Carlo Comin

  3. Department of Computer Science, University of Verona, Verona, Italy

    Romeo Rizzi

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  1. Carlo Comin
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  2. Romeo Rizzi
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Correspondence to Carlo Comin.

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Comin, C., Rizzi, R. Improved Pseudo-polynomial Bound for the Value Problem and Optimal Strategy Synthesis in Mean Payoff Games. Algorithmica 77, 995–1021 (2017). https://doi.org/10.1007/s00453-016-0123-1

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  • DOI: https://doi.org/10.1007/s00453-016-0123-1

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