Transforming Combinatorial Optimization Problems in Fourier Space: Consequences and Uses

We analyze three permutation-based combinatorial optimization problems in Fourier space, namely, the quadratic assignment problem, the linear ordering problem (LOP), and the symmetric and nonsymmetric traveling salesperson problem (STSP). In previous studies, one can find a number of theorems with n...

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Vydáno v:IEEE transactions on evolutionary computation Ročník 29; číslo 4; s. 977 - 989
Hlavní autoři: Elorza, Anne, Benavides, Xabier, Ceberio, Josu, Hernando, Leticia, Lozano, Jose A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: IEEE 01.08.2025
Témata:
Combinatorial optimization problems (COPs)
equivalent instances
Fourier transform (FT)
Fourier transforms
Linear programming
Optimization
permutations
Space exploration
Symmetric matrices
Urban areas
Vectors
ISSN:1089-778X, 1941-0026
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Shrnutí:We analyze three permutation-based combinatorial optimization problems in Fourier space, namely, the quadratic assignment problem, the linear ordering problem (LOP), and the symmetric and nonsymmetric traveling salesperson problem (STSP). In previous studies, one can find a number of theorems with necessary conditions that the Fourier coefficients of the aforementioned problems must satisfy. In this manuscript, we prove the sufficiency of these conditions, which implies that they constitute the exact characterization of the problems in Fourier space. In addition, the Fourier coefficients of the LOP and the symmetric and non-STSP are completely characterized by showing certain proportionality patterns that they must follow. Taking the characterization in Fourier space of the problems as a basis, we study classes of equivalent instances of the LOP and the symmetric and non-STSP, considering that two instances are equivalent if they have the same objective function. Furthermore, we give canonical representations for each problem in such a way that the input matrices have the minimum number of nonzero parameters.
ISSN:1089-778X
1941-0026
DOI:10.1109/TEVC.2024.3457268