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Bi-criteria path problem with minimum length and maximum survival probability

Author

Listed:
  • Nir Halman

    (Hebrew University of Jerusalem)

  • Mikhail Y. Kovalyov

    (National Academy of Sciences of Belarus)

  • Alain Quilliot

    (Université Blaise Pascal, (Clermont-Ferrand II, LIMOS))

  • Dvir Shabtay

    (Ben-Gurion University of the Negev)

  • Moshe Zofi

    (Sapir College)

Abstract

We study a bi-criteria path problem on a directed multigraph with cycles, where each arc is associated with two parameters. The first is the survival probability of moving along the arc, and the second is the length of the arc. We evaluate the quality of a path by two independent criteria. The first is to maximize the survival probability along the entire path, which is the product of the arc probabilities, and the second is to minimize the total path length, which is the sum of the arc lengths. We prove that the problem of finding a path which satisfies two bounds, one for each criterion, is NP-complete, even in the acyclic case. We further develop approximation algorithms for the optimization versions of the studied problem. One algorithm is based on approximate computing of logarithms of arc probabilities, and the other two are fully polynomial time approximation schemes (FPTASes). One FPTAS is based on scaling and rounding of the input, while the other FPTAS is derived via the method of K-approximation sets and functions, introduced by Halman et al. (Math Oper Res 34:674–685, 2009).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:orspec:v:41:y:2019:i:2:d:10.1007_s00291-018-0543-1
    DOI: 10.1007/s00291-018-0543-1
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    References listed on IDEAS

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    1. 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.
    2. Nir Halman, 2020. "A technical note: fully polynomial time approximation schemes for minimizing the makespan of deteriorating jobs with nonlinear processing times," Journal of Scheduling, Springer, vol. 23(6), pages 643-648, December.

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