Lower bounds for the universal TSP on the plane

TL;DR

This paper establishes a new lower bound for the universal TSP heuristic on the plane, showing that for any linear order, there are large point sets where the heuristic's path is significantly longer than the optimal, improving previous bounds.

Contribution

It provides an improved lower bound on the performance of the universal TSP heuristic on the plane, demonstrating inherent limitations of any linear ordering approach.

Findings

Lower bound of $C \, \sqrt{\log |S| / \log \log |S|}$ for the heuristic's path length

Dichotomy in long walks on a cycle: zig-zagging or staying within small diameter

Improvement over previous lower bound of $C \, \sqrt[6]{\log |S| / \log \log |S|}$

Abstract

We show a lower bound for the universal traveling salesman heuristic on the plane: for any linear order on the unit square $[0,1]^2$, there are finite subsets $S \subset [0,1]^2$ of arbitrarily large size such that the path visiting each element of $S$ according to the linear order has length $\geq C \sqrt{\log |S| / \log \log |S|}$ times the length of the shortest path visiting each element in $S$. ($C>0$ is a constant that depends only on the linear order.) This improves the previous lower bound $\geq C \sqrt[6]{\log |S| / \log \log |S|}$ of [HKL06]. The proof establishes a dichotomy about any long walk on a cycle: the walk either zig-zags between two far away points, or else for a large amount of time it stays inside a set of small diameter.