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

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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).

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Authors and Affiliations

  1. Hebrew University of Jerusalem, Jerusalem, Israel

    Nir Halman

  2. United Institute of Informatics Problems, National Academy of Sciences of Belarus, Minsk, Belarus

    Mikhail Y. Kovalyov

  3. Université Blaise Pascal, (Clermont-Ferrand II, LIMOS), Aubiére Cedex, France

    Alain Quilliot

  4. Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Dvir Shabtay

  5. Department of Industrial Management, Sapir College, Sderot, Israel

    Moshe Zofi

Authors
  1. Nir Halman
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  2. Mikhail Y. Kovalyov
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  3. Alain Quilliot
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  4. Dvir Shabtay
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  5. Moshe Zofi
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Correspondence to Dvir Shabtay.

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Halman, N., Kovalyov, M.Y., Quilliot, A. et al. Bi-criteria path problem with minimum length and maximum survival probability. OR Spectrum 41, 469–489 (2019). https://doi.org/10.1007/s00291-018-0543-1

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