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

Abstract

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.

For points in the plane Platzman and Bartholdi [J. ACM, 36(4):719–737, 1989] achieved a competitive ratio of $O(\log n)$ using an ordering derived from the Sierpinski curve. We introduce the notion of hierarchical orderings which captures all the commonly discussed orderings for the UTSP, including those derived from the Sierpinski, Hilbert, Lebesgue and Peano curves.

Our main result is a lower bound of $\varOmega (\log n)$ on the competitive ratio of any Universal TSP tour using hierachical orderings. This is an improvement for this setting on the best known lower bound for Universal TSP on the plane for arbitrary orderings of $\varOmega \left( \root 6 \of {\frac{\log n}{\log \log n}}\, \right)$ due to Hajiaghayi et al. [Proc. of SODA, 649–658, 2006].