A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

TL;DR

This paper presents an improved approximation scheme for Euclidean TSP that achieves near-optimal dependence on the approximation parameter, using a novel sparsity-sensitive technique based on quadtree methods.

Contribution

It introduces a new algorithm with optimal dependence on , improving previous bounds, and extends the approach to Steiner Tree problems, supported by matching lower bounds.

Findings

New algorithm with ^{d-1} dependence on 

Extension of techniques to Steiner Tree and Rectilinear Steiner Tree

Matching lower bounds under Gap-ETH hypothesis

Abstract

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ of the algorithm by Rao and Smith~(STOC 1998). We also show that a $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. We demonstrate that our technique extends to other problems, by showing that for Steiner Tree and Rectilinear Steiner Tree it yields the same running time. We complement our results with a matching Gap-ETH lower bound for Rectilinear Steiner Tree.