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A Linear Time Gap-ETH-Tight Approximation Scheme for Euclidean TSP

Tobias Mömke, Hang Zhou

TL;DR

A randomized $2^{O(1/\varepsilon^{d-1})} n$ time approximation scheme for Euclidean TSP is given, achieving a Gap-ETH tight dependence on $\varepsilon$.

Abstract

The Traveling Salesman Problem (TSP) in the $d$-dimensional Euclidean space is among the oldest and most famous NP-hard optimization problems. In breakthrough works, Arora [J. ACM 1998] and Mitchell [SICOMP 1999] gave the first polynomial time approximation schemes. To improve the running time, Rao and Smith [STOC 1998] gave a randomized $(1/\varepsilon)^{O(1/\varepsilon^{d-1})}\cdot n\log n$ time approximation scheme. Bartal and Gottlieb [FOCS 2013] gave a randomized approximation scheme in $2^{(1/\varepsilon)^{O(d)}} n$ time, which is linear in $n$. Recently, Kisfaludi-Bak, Nederlof, and Węgrzycki [FOCS 2021] gave a randomized approximation scheme in $2^{O(1/\varepsilon^{d-1})} n \log n$ time, achieving a Gap-ETH tight dependence on $\varepsilon$. It is raised as a challenging open question by Kisfaludi-Bak, Nederlof, and Węgrzycki [FOCS 2021] whether a running time of $2^{O(1/\varepsilon^{d-1})}n$ is achievable. We answer their question positively by giving a randomized $2^{O(1/\varepsilon^{d-1})} n$ time approximation scheme for Euclidean TSP.

A Linear Time Gap-ETH-Tight Approximation Scheme for Euclidean TSP

TL;DR

A randomized time approximation scheme for Euclidean TSP is given, achieving a Gap-ETH tight dependence on .

Abstract

The Traveling Salesman Problem (TSP) in the -dimensional Euclidean space is among the oldest and most famous NP-hard optimization problems. In breakthrough works, Arora [J. ACM 1998] and Mitchell [SICOMP 1999] gave the first polynomial time approximation schemes. To improve the running time, Rao and Smith [STOC 1998] gave a randomized time approximation scheme. Bartal and Gottlieb [FOCS 2013] gave a randomized approximation scheme in time, which is linear in . Recently, Kisfaludi-Bak, Nederlof, and Węgrzycki [FOCS 2021] gave a randomized approximation scheme in time, achieving a Gap-ETH tight dependence on . It is raised as a challenging open question by Kisfaludi-Bak, Nederlof, and Węgrzycki [FOCS 2021] whether a running time of is achievable. We answer their question positively by giving a randomized time approximation scheme for Euclidean TSP.

Paper Structure

This paper contains 30 sections, 19 theorems, 25 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Notations and Previous Techniques
  4. Dissection and Quadtree
  5. Approach of Arora
  6. Approach of Kisfaludi-Bak, Nederlof, and Węgrzycki
  7. Linear-time 2-approximation by Bartal and Gottlieb
  8. Portals
  9. Overview of Our Methods
  10. r-basic Tours
  11. Structure Theorem
  12. Algorithm
  13. Proof of the Structure Theorem in Two Dimensions
  14. Construction of $\mathrm{OPT}'$.
  15. Multiple Crossings With a Side: Previous Analysis
  16. ...and 15 more sections

Key Result

Theorem 1

There is a randomized $(1+\varepsilon)$-approximation scheme for the Euclidean TSP in $\mathbb{R}^d$ that runs in time $2^{O(1/\varepsilon^{d-1})}n$ in the real-RAM model with atomic floor or mod operations.

Figures (9)

  • Figure 1: The edge $e$ crosses the side $F_e$. Here $C_1$ is the rectangle consisting of the two subcells to the left of $F_e$, and $C_2$ is the rectangle consisting of the two subcells to the right of $F_e$.
  • Figure 2: The dashed red lines depict sub-paths of $T$.
  • Figure 3: The edge $e$ is the segment in green. $F_e$ is the vertical segment in bold. $F$ is the horizontal segment in bold. $A$ is the intersection point between $F_e$ and $F$. $B$ is the intersection point between $e$ and $F$. Then $\mathrm{dist}(A,B)\leq \mathrm{cost}(e)$.
  • Figure 4: The cell $C$ is represented by the cube. $F_e$ is the highlighted square, which is a mid-square of $C$. The edge $e=(u,v)$ is in green. $\pi_1$ and $\pi_2$ are the red dashed curves. $z$ is the crossing point between $e$ and $F_e$.
  • Figure 5: Three rectangles in \ref{['lem:3d-2crossings']}.
  • ...and 4 more figures

Theorems & Definitions (39)

  • Theorem 1: main theorem
  • Remark
  • Lemma 2: Arora's Patching Lemma Aro98, see also KNW21
  • Lemma 3: Bartal and Gottlieb BG13
  • Definition 4: grid, KNW21
  • Definition 5: adaptation from KNW21
  • Definition 6
  • Theorem 7: Structure Theorem
  • Definition 8
  • Definition 9: $r$-basic
  • ...and 29 more