Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Sándor Kisfaludi-Bak, Jesper Nederlof, Karol Węgrzycki

TL;DR

This paper resolves the epsilon-exponential dependence bottleneck in approximation schemes for Euclidean TSP in fixed dimensions by introducing Sparsity-Sensitive Patching, which adapts patching granularity to local sparsity and yields a Gap-ETH-tight $(1+\varepsilon)$-approximation running in $2^{\mathcal{O}(1/\varepsilon^{d-1})} \cdot n + \mathrm{poly}(1/\varepsilon) \cdot n \log n$. The approach builds on Arora's quadtree framework but avoids iterative bottom-up patching, enabling a clean rank-based DP over portals with a representative set of size $2^{\mathcal{O}(|B|)}$. The authors extend these techniques to Rectilinear and Euclidean Steiner Tree, providing matching Gap-ETH lower bounds for the rectilinear variant and outlining reductions from Max-(3,3)SAT. Overall, the work conditionalizes the optimality of the epsilon-exponent in approximation schemes for Euclidean TSP in constant dimensions and broadens the toolkit for geometric optimization problems with near-optimal, dimension-robust running times.

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.

A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

TL;DR

This paper resolves the epsilon-exponential dependence bottleneck in approximation schemes for Euclidean TSP in fixed dimensions by introducing Sparsity-Sensitive Patching, which adapts patching granularity to local sparsity and yields a Gap-ETH-tight -approximation running in . The approach builds on Arora's quadtree framework but avoids iterative bottom-up patching, enabling a clean rank-based DP over portals with a representative set of size . The authors extend these techniques to Rectilinear and Euclidean Steiner Tree, providing matching Gap-ETH lower bounds for the rectilinear variant and outlining reductions from Max-(3,3)SAT. Overall, the work conditionalizes the optimality of the epsilon-exponent in approximation schemes for Euclidean TSP in constant dimensions and broadens the toolkit for geometric optimization problems with near-optimal, dimension-robust running times.

Abstract

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). 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.

Paper Structure

This paper contains 48 sections, 32 theorems, 45 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. The existing approximation schemes and their limitations
  4. Why do known techniques fail to get a better running time?
  5. Our technique: Sparsity-Sensitive Patching
  6. Bounding the patching cost.
  7. More related work
  8. Organization
  9. Preliminaries
  10. Dissection and Quadtree.
  11. Building blocks of Arora's technique
  12. Structure Theorem
  13. Sketch for the 2-dimensional case
  14. Sketch of the algorithm
  15. Sketch of the patching
  16. ...and 33 more sections

Key Result

Theorem 1.1

For any integer $d\geqslant 2$, there is a randomized $(1+\varepsilon)$-approximation scheme for Euclidean TSP in $\mathbb{R}^d$ that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})}n + \mathrm{poly}(1/\varepsilon) n \log(n)$ time. Moreover, this cannot be improved to a $2^{o(1/\varepsilon^{d-1})}\cdot

Figures (7)

  • Figure 1: On neighboring cells of the quadtree, one must ensure that the tour crosses at most $1/\varepsilon$ times, chosen from a limited set of portals. Left: Arora's structure theorem snaps the tour to one of $\mathcal{O}(\frac{\log n}{\varepsilon})$ equally spaced portals. Right: The number of possible portal locations depends on the number of crossings; the fewer portals are used, the more precisely they are chosen. Both techniques use the Patching Lemma between the bottom two cells as their shared boundary is crossed more than $1/\varepsilon$ times.
  • Figure 2: Construction of a set of line segments $\textsf{PF}_F$ in $d=2$. The tour $\pi$ is colored red. Green portals denote the points in $\mathrm{grid}(F,r^2/|G|)$. The leftmost point and the points with $\mathrm{pro}(x) > L/(2^i/r)$ form the set $G$ and are connected to the closest portal from the grid by a green arrow. Points with $\mathrm{pro}(x) \leqslant L/(2^ir)$ form the set $N$ and they are connected to their parent with black arrows. The set of line segments $\textsf{PF}_F$ is indicated with a collection of black and green arrows.
  • Figure 3: Constructing the tree $T_0$. The spanning tree of the skeleton of a cell $C$ is connected to a vertex of its parent's spanning tree with a length-0 edge at $\mathrm{cor}(C)$ (denoted by a circles). The spanning trees at levels 1,2 and 3 are drawn in blue, orange and green, respectively.
  • Figure 4: Closest descendant crossings ($\mathrm{cdc}(.)$) and proximity in $T$, where crossing nodes are denoted by disks. The closest descendant crossing of each node $x\in X$ (be it a crossing or a branching) is the crossing node of the same color. The proximity of a crossing node $v\in I(\pi,h)$ is the total length of the tree path of the same color, i.e., the total length of the arcs entering $\mathrm{cdc}^{-1}(v)$, except for the dark blue crossing $\mathrm{cdc}(\rho)$ whose proximity is $\infty$.
  • Figure 5: Construction of the forests $\textsf{PF}_F$ in four faces in a plane $h$ of level $1$ in the quadtree. The green (thin and thick) edges are a schematic picture of $T$ (note that the actual edges consist only of axis-parallel segments), and the thick (red and green) edges indicate the forests $\textsf{PF}_F$.
  • ...and 2 more figures

Theorems & Definitions (74)

  • Theorem 1.1: Main result
  • Theorem 1.2
  • Lemma 2.1: c.f., Lemma 19.4.1 in geomspannet
  • Lemma 2.2: Patching Lemma Arora98
  • Lemma 2.3: Lemma 19.4.3 geomspannet
  • Definition 2.4: $m$-regular set
  • Definition 2.5: $r$-light
  • Theorem 2.6: Arora's Structure Theorem
  • Definition 3.1: $r$-simple geometric graph
  • Definition 3.2: $r$-simplification
  • ...and 64 more