Taxonomization of Combinatorial Optimization Problems in Fourier Space

TL;DR

This paper introduces a Fourier-based framework for analyzing permutation-based combinatorial optimization problems, aiming to classify problem instances by their spectral properties to improve understanding and algorithm design.

Contribution

It develops a novel Fourier transform approach to characterize and classify combinatorial problems, creating a taxonomy based on spectral similarities.

Findings

Fourier coefficients of TSP, Linear Ordering, and Quadratic Assignment Problems characterized.

Problems can be viewed in a homogeneous Fourier space.

Framework facilitates grouping similar problem instances.

Abstract

We propose and develop a novel framework for analyzing permutation-based combinatorial optimization problems, which could eventually be extended to other types of problems. Our approach is based on the decomposition of the objective functions via the generalized Fourier transform. We characterize the Fourier coefficients of three different problems: the Traveling Salesman Problem, the Linear Ordering Problem and the Quadratic Assignment Problem. This implies that these three problems can be viewed in a homogeneous space, such as the Fourier domain. Our final target would be to create a taxonomy of problem instances, so that functions which are treated similarly under the same search algorithms are grouped together. For this purpose, we simplify the representations of the objective functions by considering them as permutations of the elements of the search space, and study the permutations that are associated with different problems.