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 Hajiaghayi, Kleinberg and Leighton (SODA 2006). 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.

翻译：本文给出了平面上通用旅行商启发式算法的一个下界：对于单位正方形$[0,1]^2$上的任意线性序，存在任意大的有限子集$S \subset [0,1]^2$，使得按照该线性序依次访问$S$中每个元素的路径长度，至少为访问$S$中所有元素的最短路径长度的$C \sqrt{\log |S| / \log \log |S|}$倍（其中$C>0$为该线性序决定的常数）。这改进了Hajiaghayi、Kleinberg和Leighton（SODA 2006）先前得到的下界$\geq C \sqrt[6]{\log |S| / \log \log |S|}$。证明过程建立了关于环上任意长游走的一个二分性质：该游走要么在两个相距较远的点之间来回振荡，要么在大部分时间内停留在一个直径较小的集合内。