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Strong polynomiality of the Gass-Saaty shadow-vertex pivoting rule for controlled random walks

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  • Guy Even
  • Alexander Zadorojniy

Abstract

We consider the subclass of linear programs that formulate Markov Decision Processes ( mdps). We show that the Simplex algorithm with the Gass-Saaty shadow-vertex pivoting rule is strongly polynomial for a subclass of mdps, called controlled random walks (CRWs); the running time is O(|S| 3 ⋅|U| 2 ), where |S| denotes the number of states and |U| denotes the number of actions per state. This result improves the running time of Zadorojniy et al. (Mathematics of Operations Research 34(4):992–1007, 2009 ) algorithm by a factor of |S|. In particular, the number of iterations needed by the Simplex algorithm for CRWs is linear in the number of states and does not depend on the discount factor. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Guy Even & Alexander Zadorojniy, 2012. "Strong polynomiality of the Gass-Saaty shadow-vertex pivoting rule for controlled random walks," Annals of Operations Research, Springer, vol. 201(1), pages 159-167, December.
    • Handle: RePEc:spr:annopr:v:201:y:2012:i:1:p:159-167:10.1007/s10479-012-1199-x
    DOI: 10.1007/s10479-012-1199-x
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    References listed on IDEAS

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    1. Richard F. Serfozo, 1979. "Technical Note—An Equivalence Between Continuous and Discrete Time Markov Decision Processes," Operations Research, INFORMS, vol. 27(3), pages 616-620, June.
    2. M. Yadin & P. Naor, 1967. "On queueing systems with variable service capacities," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(1), pages 43-53.
    3. Alexander Zadorojniy & Guy Even & Adam Shwartz, 2009. "A Strongly Polynomial Algorithm for Controlled Queues," Mathematics of Operations Research, INFORMS, vol. 34(4), pages 992-1007, November.
    4. Alan S. Manne, 1960. "Linear Programming and Sequential Decisions," Management Science, INFORMS, vol. 6(3), pages 259-267, April.
    5. Éva Tardos, 1986. "A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs," Operations Research, INFORMS, vol. 34(2), pages 250-256, April.
    6. Karl-Heinz Borgwardt, 1982. "Some Distribution-Independent Results About the Asymptotic Order of the Average Number of Pivot Steps of the Simplex Method," Mathematics of Operations Research, INFORMS, vol. 7(3), pages 441-462, August.
    7. Cyrus Derman, 1962. "On Sequential Decisions and Markov Chains," Management Science, INFORMS, vol. 9(1), pages 16-24, October.
    8. Yinyu Ye, 2011. "The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 593-603, November.
    9. Yinyu Ye, 2005. "A New Complexity Result on Solving the Markov Decision Problem," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 733-749, August.
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