An Optimal Lower Bound for Hierarchical Universal Solutions for TSP on the Plane

A Universal TSP tour on a metric space is a total order defined over all points in the space, such that an approximate traveling salesman tour on any finite subset can be found by visiting each point of the subset in the induced order. The performance of a UTSP tour is evaluated by comparing the worst-case ratio of the length of the induced tour to the length of the optimal TSP tour over all subsets of size n. This problem has attracted significant interest over the past thirty years, especially in the case where the locations are points in the Euclidean plane.