TY - JOUR

T1 - Bi-criteria path problem with minimum length and maximum survival probability

AU - Halman, Nir

AU - Kovalyov, Mikhail Y.

AU - Quilliot, Alain

AU - Shabtay, Dvir

AU - Zofi, Moshe

N1 - Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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).

AB - 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).

KW - Approximation algorithms

KW - Bi-criteria optimization

KW - Shortest path problem

KW - Survival probability

UR - http://www.scopus.com/inward/record.url?scp=85058217448&partnerID=8YFLogxK

U2 - 10.1007/s00291-018-0543-1

DO - 10.1007/s00291-018-0543-1

M3 - Article

AN - SCOPUS:85058217448

VL - 41

SP - 469

EP - 489

JO - Operations-Research-Spektrum

JF - Operations-Research-Spektrum

SN - 0171-6468

IS - 2

ER -