IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v45y2020i1p34-62.html
   My bibliography  Save this article

Polynomial Time Algorithms for Branching Markov Decision Processes and Probabilistic Min(Max) Polynomial Bellman Equations

Author

Listed:
  • Kousha Etessami

    (School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom;)

  • Alistair Stewart

    (Department of Computer Science, University of Southern California, Los Angeles, California 90089;)

  • Mihalis Yannakakis

    (Department of Computer Science, Columbia University, New York, New York 10027)

Abstract

We show that one can compute the least nonnegative solution (also known as the least fixed point ) for a system of probabilistic min (max) polynomial equations, to any desired accuracy ɛ > 0 in time polynomial in both the encoding size of the system and in log(1/ ɛ ). These are Bellman optimality equations for important classes of infinite-state Markov decision processes (MDPs), including branching MDPs (BMDPs), which generalize classic multitype branching stochastic processes. We thus obtain the first polynomial time algorithm for computing, to any desired precision, optimal (maximum and minimum) extinction probabilities for BMDPs. Our algorithms are based on a novel generalization of Newton’s method, which employs linear programming in each iteration. We also provide polynomial-time (P-time) algorithms for computing an ɛ -optimal policy for both maximizing and minimizing extinction probabilities in a BMDP, whereas we note a hardness result for computing an exact optimal policy. Furthermore, improving on prior results, we provide more efficient P-time algorithms for qualitative analysis of BMDPs, that is, for determining whether the maximum or minimum extinction probability is 1, and, if so, computing a policy that achieves this. We also observe some complexity consequences of our results for branching simple stochastic games, which generalize BMDPs.

Suggested Citation

  • Kousha Etessami & Alistair Stewart & Mihalis Yannakakis, 2020. "Polynomial Time Algorithms for Branching Markov Decision Processes and Probabilistic Min(Max) Polynomial Bellman Equations," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 34-62, February.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:1:p:34-62
    DOI: 10.1287/moor.2018.0970
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/moor.2018.0970
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2018.0970?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Uriel G. Rothblum & Peter Whittle, 1982. "Growth Optimality for Branching Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 7(4), pages 582-601, November.
    2. Stanley R. Pliska, 1976. "Optimization of Multitype Branching Processes," Management Science, INFORMS, vol. 23(2), pages 117-124, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Eugene A. Feinberg & Jefferson Huang, 2019. "On the reduction of total‐cost and average‐cost MDPs to discounted MDPs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(1), pages 38-56, February.
    2. Yiqiang Q. Zhao & W. John Braun & Wei Li, 1999. "Northwest corner and banded matrix approximations to a Markov chain," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(2), pages 187-197, March.
    3. Daniel Hernández Hernández & Diego Hernández Bustos, 2017. "Local Poisson Equations Associated with Discrete-Time Markov Control Processes," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 1-29, April.
    4. Arie Leizarowitz, 2003. "An Algorithm to Identify and Compute Average Optimal Policies in Multichain Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 553-586, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormoor:v:45:y:2020:i:1:p:34-62. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.