A Fast Quasi-Linear Heuristic for the Close-Enough Traveling Salesman Problem

Abstract

We introduce a fast, quasi-linear-time heuristic for the Close-Enough Traveling Salesman Problem (CETSP), a continuous generalization of the Euclidean TSP in which each target is a disk that must be intersected. The method adapts the pair-center clustering paradigm to circular neighborhoods: a hierarchical clustering phase merges nearby disks into proxy circles using an R*-tree for efficient spatial queries, and a construction phase incrementally expands the hierarchy into a feasible tour while maintaining and locally optimizing tour points. Lightweight local improvements, selective reinsertion and constrained point reoptimization, reduce local inefficiencies without compromising scalability. The algorithm runs in expected O(n log n) time and, on benchmark instances reconstructed from the Mennell dataset, produces solutions within roughly 0-2% of state-of-the-art best-known values while requiring orders-of-magnitude less runtime than population-based metaheuristics. The approach trades some final-solution optimality for dramatic gains in speed and scalability, making it suitable for very large CETSP instances.

Figures (8)

• Figure 1: Visualization of the clustering phase for a sample instance with $n=9$ disks. Each subfigure shows one merge step, where the two closest circles (highlighted) are replaced by their proxy circle (dashed blue outline). The final frame (bottom right) shows the single remaining cluster.
• Figure 2: Visualization of the construction phase for the same sample instance. The process starts from the final proxy circle obtained in the clustering phase (top left) and iteratively expands proxy circles into their generating circles while updating the tour. Newly inserted circles and tour points are highlighted, while removed proxy elements are shown in a faded, dashed style. Note that we do not show point optimizations here.
• Figure 3: Approximate solution to the Alhazen problem for placing a new tour point within a circle. The bisection of angle $\angle AOB$ is used as a fast approximation of the optimal insertion point LeiHao2024.
• Figure 4: Illustration of the point reoptimization step. The new position $P^\star$ is obtained by projecting the current point $P$ toward the gradient direction of decreasing tour cost, constrained to remain within all corresponding circles.
• Figure 5: Comparison of the benchmark and our solver's tour for the bubbles9 instance. The benchmark exhibits a highly structured, spiral traversal through the concentric layers of circles, whereas our solver's path shows local inefficiencies and incomplete alignment along the optimal spiral pattern. This case highlights the solver's difficulty escaping local minima in structured overlapping geometries.