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

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...

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Bibliographic Details
Published in:OR Spectrum Vol. 41; no. 2; pp. 469 - 489
Main Authors: Halman, Nir, Kovalyov, Mikhail Y., Quilliot, Alain, Shabtay, Dvir, Zofi, Moshe
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2019
Springer Nature B.V
Subjects:
Algorithms
Approximation
Business and Management
Calculus of Variations and Optimal Control; Optimization
Criteria
Logarithms
Mathematical analysis
Operations Research/Decision Theory
Polynomials
Probability
Regular Article
Rounding
Survival
Shortest path problem
Bi-criteria optimization
Approximation algorithms
Survival probability
ISSN:0171-6468, 1436-6304
Online Access:Get full text
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Summary: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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ISSN:0171-6468
1436-6304
DOI:10.1007/s00291-018-0543-1