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Complexity of discrete multicriteria problems

Complexity of discrete multicriteria problems -- This review contains the results which concern the estimates of the computational complexity of combinatorial problems of vector optimization, the solvability of these problems in the class of algorithms of linear convolution, and the accuracy evaluation of fast algorithms of finding the set of alternatives in typical cases. The discrete, multicriteria problems considered here include polynomially solvable, polynomially reducible to the NP class, and intractable problems, i.e., they cover the main part of the scale of estimates of the computational complexity. 1. FORMULATION OF THE PROBLEMS A mathematical setting of any multicriteria problem presupposed that a set X = {x} of admissible solutions (AS's) is given and a multiobjective function (MF) F(x) = (ii(*), F2(x), . . . , FN(x)) (1.1) is defined on this set. For definiteness, we assume that all the criteria of this MF should be minimized, i.e., fXaO-nnin, i/ = l,...,7V. (1.2) An admissible solution (further the word admissible will be omitted) G X is said to be a Pareto optimal solution or a Pareto optimum (PO) if there is no x" £ X such that F(x~) < F(x)9 F(x*) £ F(x) (see [1]). The symbol X denotes the Pareto set consisting of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

Complexity of discrete multicriteria problems

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References (21)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1994.4.2.89
Publisher site
See Article on Publisher Site

Abstract

-- This review contains the results which concern the estimates of the computational complexity of combinatorial problems of vector optimization, the solvability of these problems in the class of algorithms of linear convolution, and the accuracy evaluation of fast algorithms of finding the set of alternatives in typical cases. The discrete, multicriteria problems considered here include polynomially solvable, polynomially reducible to the NP class, and intractable problems, i.e., they cover the main part of the scale of estimates of the computational complexity. 1. FORMULATION OF THE PROBLEMS A mathematical setting of any multicriteria problem presupposed that a set X = {x} of admissible solutions (AS's) is given and a multiobjective function (MF) F(x) = (ii(*), F2(x), . . . , FN(x)) (1.1) is defined on this set. For definiteness, we assume that all the criteria of this MF should be minimized, i.e., fXaO-nnin, i/ = l,...,7V. (1.2) An admissible solution (further the word admissible will be omitted) G X is said to be a Pareto optimal solution or a Pareto optimum (PO) if there is no x" £ X such that F(x~) < F(x)9 F(x*) £ F(x) (see [1]). The symbol X denotes the Pareto set consisting of

Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1994

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