An Adaptive Layered Clustering Framework with Improved Genetic Algorithm for Solving Large-Scale Traveling Salesman Problems

Abstract

Traveling salesman problems (TSPs) are well-known combinatorial optimization problems, and most existing algorithms are challenging for solving TSPs when its scale is large. To improve the efficiency of solving large-scale TSPs, this work presents a novel adaptive layered clustering framework with improved genetic algorithm (ALC\_IGA). The primary idea behind ALC\_IGA is to break down a large-scale problem into a series of small-scale problems. First, the $k$-means and improved genetic algorithm are used to segment the large-scale TSPs layer by layer and generate the initial solution. Then, the developed two phases simplified $2$-opt algorithm is applied to further improve the quality of the initial solution. The analysis reveals that the computational complexity of the ALC\_IGA is between $O(n\log n)$ and $O(n^2)$. The results of numerical experiments on various TSP instances indicate that, in most situations, the ALC\_IGA surpasses the state-of-the-art algorithms in convergence speed, stability, and solution quality. Specifically, the ALC\_IGA can solve instances with $2 \times 10^5$ nodes within 0.15h, $1.4 \times 10^6$ nodes within 1h, and $2 \times 10^6$ nodes in three dimensions within 1.5h.