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arXiv · 2412.16448

Lower bounds for the universal TSP on the plane

Cosmas Kravaris
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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 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.

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Cosmas Kravaris. 2024-12-21. Lower bounds for the universal TSP on the plane. https://arxiv.org/abs/2412.16448

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