In this paper, we consider approximation algorithms for Euclidean TSP problem.

What is it about?

We revisit the classic task of finding the shortest tour of points in -dimensional Euclidean space, for any fixed constant . We determine the optimal dependence on in the running time of an algorithm that computes a -approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in time. This improves the previously smallest dependence on in the running time of the algorithm by Rao and Smith (STOC 1998). We also show that a algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH).

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Photo by Rehan Syed on Unsplash

Photo by Rehan Syed on Unsplash

Why is it important?

Our algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call sparsity-sensitive patching. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally.