A tabu search with geometry-based sparsification methods for angular traveling salesman problems

The angular-metric traveling salesman problem (AngleTSP) aims to find a tour visiting a given set of vertices in the Euclidean plane exactly once while minimizing the cost given by the sum of all turning angles. If the cost is obtained by combining the sum of all turning angles and the traveled distance, the problem is called angular-distance-metric traveling salesman problem (AngleDistanceTSP). In this work, we study the symmetric variants of these problems. Because both the AngleTSP and AngleDistanceTSP are NP-hard, multiple heuristic approaches have been proposed in the literature. Nevertheless, a good tradeoff between solution quality and runtime is hard to find. We propose a granular tabu search (GTS) that considers the geometric features of the two problems in the design of starting solutions and sparsification methods. We further enrich the GTS with components that guarantee both intensification and diversification during the search. The computational results on benchmark instances from the literature show that (i) for the AngleTSP, our GTS lies on the Pareto frontier of the best performing-heuristics, and (ii) for the AngleDistanceTSP, our GTS provides the best solution quality across all existing heuristics in competitive runtimes. In addition, new best-known solutions are found for most benchmark instances for which an optimal solution is not available.