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

TL;DR

This paper presents a fast, scalable heuristic for the CETSP that achieves near-optimal solutions in expected O(n log n) time, significantly outperforming existing metaheuristics in speed.

Contribution

A novel quasi-linear-time heuristic for CETSP that combines hierarchical clustering with local optimization, enabling efficient solutions for large instances.

Findings

Runs in expected O(n log n) time.

Achieves solutions within 0-2% of best-known values.

Requires orders-of-magnitude less runtime than metaheuristics.

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.