IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v300y2022i1p35-45.html
   My bibliography  Save this article

A fully polynomial time approximation scheme for the probability maximizing shortest path problem

Author

Listed:
  • Lee, Jisun
  • Joung, Seulgi
  • Lee, Kyungsik

Abstract

In this paper, we consider a probability maximizing shortest path problem. For a given directed graph with the length of each arc being an independent normal random variable with rational mean and standard deviation, the problem is to find a simple s−t path that maximizes the probability of arriving at the destination within a given limit. We first prove that the problem is NP-hard even on directed acyclic graphs with the mean of the length of each arc being restricted to an integer. Then, we present pseudo-polynomial time exact algorithms for the problem along with nontrivial special cases that can be solved in polynomial time. Finally, we present a fully polynomial time approximation scheme (FPTAS) that iteratively solves deterministic shortest path problems. The structure of the proposed approximation scheme can be applied to devise FPTAS for other probability maximizing combinatorial optimization problems once the corresponding deterministic problems are polynomially solvable.

Suggested Citation

  • Lee, Jisun & Joung, Seulgi & Lee, Kyungsik, 2022. "A fully polynomial time approximation scheme for the probability maximizing shortest path problem," European Journal of Operational Research, Elsevier, vol. 300(1), pages 35-45.
    • Handle: RePEc:eee:ejores:v:300:y:2022:i:1:p:35-45
    DOI: 10.1016/j.ejor.2021.10.018
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221721008638
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2021.10.018?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. Chassein, André B. & Goerigk, Marc, 2015. "A new bound for the midpoint solution in minmax regret optimization with an application to the robust shortest path problem," European Journal of Operational Research, Elsevier, vol. 244(3), pages 739-747.
    2. Taccari, Leonardo, 2016. "Integer programming formulations for the elementary shortest path problem," European Journal of Operational Research, Elsevier, vol. 252(1), pages 122-130.
    3. Guillot, Matthieu & Stauffer, Gautier, 2020. "The Stochastic Shortest Path Problem: A polyhedral combinatorics perspective," European Journal of Operational Research, Elsevier, vol. 285(1), pages 148-158.
    4. Murthy, Ishwar & Sarkar, Sumit, 1997. "Exact algorithms for the stochastic shortest path problem with a decreasing deadline utility function," European Journal of Operational Research, Elsevier, vol. 103(1), pages 209-229, November.
    5. Wang, Li & Yang, Lixing & Gao, Ziyou, 2016. "The constrained shortest path problem with stochastic correlated link travel times," European Journal of Operational Research, Elsevier, vol. 255(1), pages 43-57.
    6. Nir Halman & Mikhail Y. Kovalyov & Alain Quilliot & Dvir Shabtay & Moshe Zofi, 2019. "Bi-criteria path problem with minimum length and maximum survival probability," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 41(2), pages 469-489, June.
    7. Azaron, Amir & Kianfar, Farhad, 2003. "Dynamic shortest path in stochastic dynamic networks: Ship routing problem," European Journal of Operational Research, Elsevier, vol. 144(1), pages 138-156, January.
    8. Chassein, André & Dokka, Trivikram & Goerigk, Marc, 2019. "Algorithms and uncertainty sets for data-driven robust shortest path problems," European Journal of Operational Research, Elsevier, vol. 274(2), pages 671-686.
    9. Henig, Mordechai I., 1986. "The shortest path problem with two objective functions," European Journal of Operational Research, Elsevier, vol. 25(2), pages 281-291, May.
    10. Mohammad Hessam Olya, 2014. "Applying Dijkstra's algorithm for general shortest path problem with normal probability distribution arc length," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 21(2), pages 143-154.
    11. Goldberg, Noam & Poss, Michael, 2020. "Maximum probabilistic all-or-nothing paths," European Journal of Operational Research, Elsevier, vol. 283(1), pages 279-289.
    12. C. Elliott Sigal & A. Alan B. Pritsker & James J. Solberg, 1980. "The Stochastic Shortest Route Problem," Operations Research, INFORMS, vol. 28(5), pages 1122-1129, October.
    13. Wang, Zhuolin & You, Keyou & Song, Shiji & Zhang, Yuli, 2020. "Wasserstein distributionally robust shortest path problem," European Journal of Operational Research, Elsevier, vol. 284(1), pages 31-43.
    14. Raith, Andrea & Schmidt, Marie & Schöbel, Anita & Thom, Lisa, 2018. "Multi-objective minmax robust combinatorial optimization with cardinality-constrained uncertainty," European Journal of Operational Research, Elsevier, vol. 267(2), pages 628-642.
    15. George B. Dantzig, 1960. "On the Shortest Route Through a Network," Management Science, INFORMS, vol. 6(2), pages 187-190, January.
    16. Nie, Yu (Marco) & Wu, Xing, 2009. "Shortest path problem considering on-time arrival probability," Transportation Research Part B: Methodological, Elsevier, vol. 43(6), pages 597-613, July.
    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. Sergey S. Ketkov, 2023. "On the Multistage Shortest Path Problem Under Distributional Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 277-308, April.
    2. Häme, Lauri & Hakula, Harri, 2013. "Dynamic journeying under uncertainty," European Journal of Operational Research, Elsevier, vol. 225(3), pages 455-471.
    3. David Corredor-Montenegro & Nicolás Cabrera & Raha Akhavan-Tabatabaei & Andrés L. Medaglia, 2021. "On the shortest $$\alpha$$ α -reliable path problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 287-318, April.
    4. Huang, He & Gao, Song, 2012. "Optimal paths in dynamic networks with dependent random link travel times," Transportation Research Part B: Methodological, Elsevier, vol. 46(5), pages 579-598.
    5. Michael Redmond & Ann Melissa Campbell & Jan Fabian Ehmke, 2020. "Data-driven planning of reliable itineraries in multi-modal transit networks," Public Transport, Springer, vol. 12(1), pages 171-205, March.
    6. Marie Schmidt & Leo Kroon & Anita Schöbel & Paul Bouman, 2017. "The Travelers Route Choice Problem Under Uncertainty: Dominance Relations Between Strategies," Operations Research, INFORMS, vol. 65(1), pages 184-199, February.
    7. Shen, Liang & Shao, Hu & Wu, Ting & Fainman, Emily Zhu & Lam, William H.K., 2020. "Finding the reliable shortest path with correlated link travel times in signalized traffic networks under uncertainty," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 144(C).
    8. Zhang, Yuli & Shen, Zuo-Jun Max & Song, Shiji, 2016. "Parametric search for the bi-attribute concave shortest path problem," Transportation Research Part B: Methodological, Elsevier, vol. 94(C), pages 150-168.
    9. Jian Li & Amol Deshpande, 2019. "Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 354-375, February.
    10. Wang, Li & Yang, Lixing & Gao, Ziyou, 2016. "The constrained shortest path problem with stochastic correlated link travel times," European Journal of Operational Research, Elsevier, vol. 255(1), pages 43-57.
    11. Elías Escobar-Gómez & J.L. Camas-Anzueto & Sabino Velázquez-Trujillo & Héctor Hernández-de-León & Rubén Grajales-Coutiño & Eduardo Chandomí-Castellanos & Héctor Guerra-Crespo, 2019. "A Linear Programming Model with Fuzzy Arc for Route Optimization in the Urban Road Network," Sustainability, MDPI, vol. 11(23), pages 1-18, November.
    12. Xing, Tao & Zhou, Xuesong, 2011. "Finding the most reliable path with and without link travel time correlation: A Lagrangian substitution based approach," Transportation Research Part B: Methodological, Elsevier, vol. 45(10), pages 1660-1679.
    13. Zhouchun Huang & Qipeng P. Zheng & Eduardo L. Pasiliao & Daniel Simmons, 2017. "Exact algorithms on reliable routing problems under uncertain topology using aggregation techniques for exponentially many scenarios," Annals of Operations Research, Springer, vol. 249(1), pages 141-162, February.
    14. Marie Schmidt & Leo Kroon & Anita Schöbel & Paul Bouman, 2017. "The Travelers Route Choice Problem Under Uncertainty: Dominance Relations Between Strategies," Operations Research, INFORMS, vol. 65(1), pages 184-199, February.
    15. Chassein, André & Dokka, Trivikram & Goerigk, Marc, 2019. "Algorithms and uncertainty sets for data-driven robust shortest path problems," European Journal of Operational Research, Elsevier, vol. 274(2), pages 671-686.
    16. Bi Chen & William Lam & Agachai Sumalee & Qingquan Li & Hu Shao & Zhixiang Fang, 2013. "Finding Reliable Shortest Paths in Road Networks Under Uncertainty," Networks and Spatial Economics, Springer, vol. 13(2), pages 123-148, June.
    17. Matthias Ruß & Gunther Gust & Dirk Neumann, 2021. "The Constrained Reliable Shortest Path Problem in Stochastic Time-Dependent Networks," Operations Research, INFORMS, vol. 69(3), pages 709-726, May.
    18. Zohreh Hosseini Nodeh & Ali Babapour Azar & Rashed Khanjani Shiraz & Salman Khodayifar & Panos M. Pardalos, 2020. "Joint chance constrained shortest path problem with Copula theory," Journal of Combinatorial Optimization, Springer, vol. 40(1), pages 110-140, July.
    19. Chassein, André & Goerigk, Marc, 2018. "Variable-sized uncertainty and inverse problems in robust optimization," European Journal of Operational Research, Elsevier, vol. 264(1), pages 17-28.
    20. Elise D. Miller-Hooks & Hani S. Mahmassani, 2000. "Least Expected Time Paths in Stochastic, Time-Varying Transportation Networks," Transportation Science, INFORMS, vol. 34(2), pages 198-215, May.

    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:eee:ejores:v:300:y:2022:i:1:p:35-45. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    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.