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Characterizing Permutation-Based Combinatorial Optimization Problems in Fourier Space.

Anne Elorza1, Leticia Hernando2, Jose A Lozano3,4

  • 1Intelligent Systems Group, Department of Computer Science and Artificial Intelligence, University of the Basque Country UPV/EHU, 20018 San Sebastián, Spain anne.elorza@ehu.eus.

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Summary
This summary is machine-generated.

Comparing combinatorial optimization problems is challenging due to diverse definitions. This study uses Fourier transforms over the symmetric group to unify representations, revealing shared properties and aiding algorithm design for problems like the Traveling Salesperson Problem.

Keywords:
Combinatorial optimization problemsFourier transformintrinsic dimensionpermutationsrepresentation theory

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Area of Science:

  • Combinatorial Optimization
  • Harmonic Analysis
  • Computational Complexity

Background:

  • Comparing diverse combinatorial optimization problems is difficult due to varied definitions (weights, flows, distances).
  • Despite differences, problem instances often share underlying properties.
  • A unified representation framework can reveal theoretical insights and new algorithms.

Purpose of the Study:

  • To propose a unified method for comparing permutation-based combinatorial optimization problems.
  • To utilize the Fourier transform over the symmetric group for homogeneous problem representation.
  • To analyze the intersection and intrinsic dimensions of different optimization problems.

Main Methods:

  • Applying the Fourier transform over the symmetric group to permutation-based combinatorial optimization problems.
  • Building upon prior work characterizing the Fourier coefficients of the Quadratic Assignment Problem.
  • Describing the Fourier coefficients for the Symmetric Traveling Salesperson Problem, Nonsymmetric Traveling Salesperson Problem, and Linear Ordering Problem.

Main Results:

  • The Fourier transform provides a common space for comparing diverse combinatorial optimization problems.
  • Characterization of Fourier coefficients for the Traveling Salesperson Problem (symmetric and nonsymmetric) and Linear Ordering Problem.
  • The transformation facilitates understanding problem intersections and bounding intrinsic dimensions.

Conclusions:

  • The Fourier transform over the symmetric group is an effective tool for unifying and comparing permutation-based combinatorial optimization problems.
  • This approach enhances understanding of problem similarities and differences.
  • The method aids in the potential discovery of new theoretical properties and algorithmic designs.