Skip to main content
SpringerLink
Log in

An exponential lower bound for Cunningham’s rule

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we give an exponential lower bound for Cunningham’s least recently considered (round-robin) rule as applied to parity games, Markov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. We do not consider here the more general unique sink orientations (USOs), which may contain cycles, and were introduced earlier by Stickney and Watson [21] and studied by Gärtner and Schurr [13], among others.

  2. \(x \equiv _2 y\) if and only if x and y are congruent mod 2.

References

  1. Aoshima, Y., Avis, D., Deering, T., Matsumoto, Y., Moriyama, S.: On the existence of hamiltonian paths for history based pivot rules on acyclic unique sink orientations of hypercubes. Discrete Appl. Math. 160(15), 2104–2115 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Avis, D., Moriyama, S.: On combinatorial properties of linear programming digraphs. In: Avis, D., Bremner, D., Deza, A. (eds.) Polyhedral Computation, CRM Proceedings and Lecture Notes 48, pp. 1–13. AMS, Providence (2009)

    Google Scholar 

  3. Bertsekas, D.: Dynamic Programming and Optimal Control, 2nd edn. Athena Scientific, Belmont (2001)

    MATH  Google Scholar 

  4. Chvátal, V.: Linear Programmming. W.H. Freeman, New York (1983)

    Google Scholar 

  5. Cunningham, W.H.: Theoretical properties of the network simplex method. Math. Oper. Res. 4(2), 196–208 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Derman, C.: Finite State Markov Decision Processes. Academic Press, London (1972)

    MATH  Google Scholar 

  7. Emerson, E., Jutla, C.: Tree automata, \(\mu \)-calculus and determinacy. In: Proceedings 32nd Symposium on Foundations of Computer Science, pp. 368–377. IEEE, San Juand (1991)

  8. Friedmann, O.: An exponential lower bound for the latest deterministic strategy iteration algorithms. Log. Methods Comput. Sci. 7(3), 8 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Friedmann, O.: A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games. In: IPCO, pp. 192–206 (2011)

  10. Friedmann, O.: A subexponential lower bound for the least recently considered rule for solving linear programs and games. In: GAMES’2012. Naples, Italy (2012)

  11. Friedmann, O.: (2013). http://files.oliverfriedmann.de/add/cunningham_exponential_parity_game.pdf

  12. Gärtner, B.: The random-facet simplex algorithm on combinatorial cubes. Random Struct. Algorithms 20(3), 353–381 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gärtner, B., Schurr, I.: Linear programming and unique sink orientations. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 749–757 (2006)

  14. Howard, R.: Dynamic Programming and Markov Processes. MIT Press, Cambridge (1960)

    MATH  Google Scholar 

  15. Kalai, G.: A subexponential randomized simplex algorithm (extended abstract). In: STOC, pp. 475–482 (1992)

  16. Klee, V., Minty, G.J.: How Good is the Simplex Algorithm? In: Shisha, O. (ed.) Inequalities III, pp. 159–175. Academic Press Inc., New York (1972)

    Google Scholar 

  17. Matousek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. In: Symposium on Computational Geometry, pp. 1–8 (1992)

  18. Matousek, J., Szabó, T.: Random edge can be exponential on abstract cubes. In: FOCS, pp. 92–100 (2004)

  19. Puri, A.: Theory of hybrid systems and discrete event systems. Ph.D. thesis, University of California, Berkeley (1995). http://www.eecs.berkeley.edu/Pubs/TechRpts/1995/2950.html

  20. Puterman, M.: Markov Decision Processes. Wiley, London (1994)

    Book  MATH  Google Scholar 

  21. Stickney, A., Watson, L.: Digraph models of bard-type algorithms for the linear complementarity problem. Math. Oper. Res. 3(4), 322–333 (1978). doi:10.1287/moor.3.4.322

    Article  MATH  MathSciNet  Google Scholar 

  22. Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proceedings of the 42th FOCS, pp. 547–555 (2001)

  23. Vöge, J., Jurdzinski, M.: A discrete strategy improvement algorithm for solving parity games. In: Proceedings of 12th International Conference on Computer Aided Verification, CAV’00, LNCS, vol. 1855, pp. 202–215. Springer, Berlin (2000)

  24. Zadeh, N.: What is the worst case behavior of the simplex algorithm. In: Polyhedral Computation, pp. 131–143. American Mathematical Society (2009, 1980)

  25. Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theoret. Comput. Sci. 158(1–2), 343–359 (1996). doi:10.1016/0304-3975(95)00188-3

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank the referees for detailed suggestions that helped us improve the original version of this paper.

Author information

Authors and Affiliations

  1. University of Munich, Munich, Germany

    Oliver Friedmann

  2. School of Informatics, Kyoto University, Kyoto, Japan

    David Avis

  3. School of Computer Science, McGill University, Montréal, QC, Canada

    David Avis

Authors
  1. David Avis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oliver Friedmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oliver Friedmann.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avis, D., Friedmann, O. An exponential lower bound for Cunningham’s rule. Math. Program. 161, 271–305 (2017). https://doi.org/10.1007/s10107-016-1008-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-016-1008-4

Keywords

Mathematics Subject Classification

Search

Navigation