IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v55y2013i3p633-639.html
   My bibliography  Save this article

Markov chains, Hamiltonian cycles and volumes of convex bodies

Author

Listed:
  • Vivek Borkar
  • Jerzy Filar

Abstract

In this note the Hamiltonian cycle problem is mapped into an infinite horizon discounted cost constrained Markov decision problem. The occupation measure based linear polytope associated with this control problem defines a convex set which either strictly contains or is equal to another convex set, depending on whether the underlying graph has a Hamiltonian cycle or not. This allows us to distinguish Hamiltonian graphs from non-Hamiltonian graphs by comparing volumes of two convex sets. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Vivek Borkar & Jerzy Filar, 2013. "Markov chains, Hamiltonian cycles and volumes of convex bodies," Journal of Global Optimization, Springer, vol. 55(3), pages 633-639, March.
    • Handle: RePEc:spr:jglopt:v:55:y:2013:i:3:p:633-639
    DOI: 10.1007/s10898-011-9819-6
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-011-9819-6
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10898-011-9819-6?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. T. P. I. Ahamed & V. S. Borkar & S. Juneja, 2006. "Adaptive Importance Sampling Technique for Markov Chains Using Stochastic Approximation," Operations Research, INFORMS, vol. 54(3), pages 489-504, June.
    2. Eugene A. Feinberg, 2000. "Constrained Discounted Markov Decision Processes and Hamiltonian Cycles," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 130-140, February.
    3. Jerzy A. Filar & Dmitry Krass, 1994. "Hamiltonian Cycles and Markov Chains," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 223-237, February.
    4. Vladimir Ejov & Jerzy A. Filar & Michael Haythorpe & Giang T. Nguyen, 2009. "Refined MDP-Based Branch-and-Fix Algorithm for the Hamiltonian Cycle Problem," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 758-768, August.
    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. Ali Eshragh & Jerzy Filar, 2011. "Hamiltonian Cycles, Random Walks, and Discounted Occupational Measures," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 258-270, May.
    2. Ali Eshragh & Jerzy A. Filar & Thomas Kalinowski & Sogol Mohammadian, 2020. "Hamiltonian Cycles and Subsets of Discounted Occupational Measures," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 713-731, May.
    3. Eugene A. Feinberg & Uriel G. Rothblum, 2012. "Splitting Randomized Stationary Policies in Total-Reward Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 129-153, February.
    4. Vladimir Ejov & Jerzy A. Filar & Michael Haythorpe & Giang T. Nguyen, 2009. "Refined MDP-Based Branch-and-Fix Algorithm for the Hamiltonian Cycle Problem," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 758-768, August.
    5. Ali Eshragh & Jerzy Filar & Michael Haythorpe, 2011. "A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem," Annals of Operations Research, Springer, vol. 189(1), pages 103-125, September.
    6. Vladimir Ejov & Jerzy A. Filar & Minh-Tuan Nguyen, 2004. "Hamiltonian Cycles and Singularly Perturbed Markov Chains," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 114-131, February.
    7. Nelly Litvak & Vladimir Ejov, 2009. "Markov Chains and Optimality of the Hamiltonian Cycle," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 71-82, February.
    8. Konstantin Avrachenkov & Ali Eshragh & Jerzy A. Filar, 2016. "On transition matrices of Markov chains corresponding to Hamiltonian cycles," Annals of Operations Research, Springer, vol. 243(1), pages 19-35, August.
    9. Jerzy A. Filar & Asghar Moeini, 2022. "Hamiltonian cycle curves in the space of discounted occupational measures," Annals of Operations Research, Springer, vol. 317(2), pages 605-622, October.
    10. Pierre L’Ecuyer & Bruno Tuffin, 2011. "Approximating zero-variance importance sampling in a reliability setting," Annals of Operations Research, Springer, vol. 189(1), pages 277-297, September.
    11. Lanlan Zhang & Xianping Guo, 2008. "Constrained continuous-time Markov decision processes with average criteria," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(2), pages 323-340, April.
    12. Kaynar, Bahar & Ridder, Ad, 2010. "The cross-entropy method with patching for rare-event simulation of large Markov chains," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1380-1397, December.
    13. Guo, Xianping & Zhang, Wenzhao, 2014. "Convergence of controlled models and finite-state approximation for discounted continuous-time Markov decision processes with constraints," European Journal of Operational Research, Elsevier, vol. 238(2), pages 486-496.
    14. Ohlmann, Jeffrey W. & Bean, James C., 2009. "Resource-constrained management of heterogeneous assets with stochastic deterioration," European Journal of Operational Research, Elsevier, vol. 199(1), pages 198-208, November.
    15. Eugene A. Feinberg, 2000. "Constrained Discounted Markov Decision Processes and Hamiltonian Cycles," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 130-140, February.
    16. Bahar Kaynar & Ad Ridder, 2009. "The Cross-Entropy Method with Patching for Rare-Event Simulation of Large Markov Chains," Tinbergen Institute Discussion Papers 09-084/4, Tinbergen Institute.
    17. Dmitry Krass & O. J. Vrieze, 2002. "Achieving Target State-Action Frequencies in Multichain Average-Reward Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 545-566, August.
    18. Hernan P. Awad & Peter W. Glynn & Reuven Y. Rubinstein, 2013. "Zero-Variance Importance Sampling Estimators for Markov Process Expectations," Mathematics of Operations Research, INFORMS, vol. 38(2), pages 358-388, May.
    19. G. Dorini & F. Pierro & D. Savic & A. Piunovskiy, 2007. "Neighbourhood Search for constructing Pareto sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 315-337, April.
    20. J. Filar & M. Haythorpe & W. Murray, 2013. "On the determinant and its derivatives of the rank-one corrected generator of a Markov chain on a graph," Journal of Global Optimization, Springer, vol. 56(4), pages 1425-1440, 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:spr:jglopt:v:55:y:2013:i:3:p:633-639. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.