Spectral Alignment as a Structured Initialization Strategy for the Quadratic Assignment Problem

This work proposes a constructive method based on spectral alignment for the Quadratic Assignment Problem (QAP). The approach combines the principles of graph spectral theory and structural alignment to efficiently extract the global properties of flow and distance matrices. The method converts the matrices into continuous latent representations via spectral decomposition, followed by a process of analytical alignment and iterative refinement by geometric diffusion. The validation occurred in 134 instances of the QAPLIB library, comparing the performance with classical constructors of the literature. The results show that the method generates high-quality initial solutions with low computational cost, obtaining an average deviation of only 2.20% in relation to the best known solutions and surpassing traditional approaches. The method exhibited remarkable scalability, resolving complex instances up to n = 256 in less than 0.2 seconds. It is concluded, therefore, that spectral alignment is a highly effective strategy for structured initialization in combinatorial optimization.