A Convex Hull Cheapest Insertion Heuristic for the Non-Euclidean TSP

The convex hull cheapest insertion heuristic is known to produce good solutions to the Traveling Salesperson Problem in Euclidean spaces, but it has never been extended to the non-Euclidean problem. This paper proposes an adaptation that uses multidimensional scaling to first project the points from a non-Euclidean space into a Euclidean equivalent space, thereby enabling the generation of a convex hull that initializes the algorithm. To evaluate the proposed algorithm, non-Euclidean spaces are created by adding separators to the Euclidean TSPLIB benchmark data-set, or by using the L1 norm as a metric. This adapted heuristic is demonstrated to outperform the commonly used Nearest Neighbor heuristic and Nearest Insertion heuristic in 88% and 99% of the cases studied, respectively. When compared with metaheuristic algorithms, the proposed heuristic's tour costs are lower than the solutions found by the genetic algorithm and ant colony optimization algorithm in 87% and 95% of the instances, respectively.