Lower bounds for the universal TSP on the plane
Cosmas Kravaris
TL;DR
The paper proves a universal lower bound for the traveling salesman problem on the plane under any linear order: for large finite point sets $S$ in $[0,1]^2$, the ordered-visit cost $cost_{\le}(S)$ cannot be too small relative to the TSP optimum $tsp(S)$. The authors develop a two-pronged obstruction framework, showing that any long walk either zig-zags between distant points or remains confined within a small-diameter region for long stretches. They establish a robust lower bound $OR_{\le}(n) \gtrsim \sqrt{\log n / \log \log n}$ by combining a new spiral-chain zig-zag construction with a backtrack-based dyadic analysis and a probabilistic random-line optimization to control the obstruction distribution. This improves the prior $\sqrt[6]{\log n / \log \log n}$ bound and advances understanding of universal TSP behavior on the plane, with potential extensions to other metric spaces and linearisable orderings.
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.