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Faster Approximation Scheme for Euclidean $k$-TSP

Ernest van Wijland, Hang Zhou

TL;DR

The algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor $O(n^d)$ by improving Arora's approximation scheme of running time.

Abstract

In the Euclidean $k$-traveling salesman problem ($k$-TSP), we are given $n$ points in the $d$-dimensional Euclidean space, for some fixed constant $d\geq 2$, and a positive integer $k$. The goal is to find a shortest tour visiting at least $k$ points. We give an approximation scheme for the Euclidean $k$-TSP in time $n\cdot 2^{O(1/\varepsilon^{d-1})} \cdot(\log n)^{2d^2\cdot 2^d}$. This improves Arora's approximation scheme of running time $n\cdot k\cdot (\log n)^{\left(O\left(\sqrt{d}/\varepsilon\right)\right)^{d-1}}$ [J. ACM 1998]. Our algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor $O(n^d)$.

Faster Approximation Scheme for Euclidean $k$-TSP

TL;DR

The algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor by improving Arora's approximation scheme of running time.

Abstract

In the Euclidean -traveling salesman problem (-TSP), we are given points in the -dimensional Euclidean space, for some fixed constant , and a positive integer . The goal is to find a shortest tour visiting at least points. We give an approximation scheme for the Euclidean -TSP in time . This improves Arora's approximation scheme of running time [J. ACM 1998]. Our algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor .
Paper Structure (15 sections, 11 theorems, 32 equations, 3 algorithms)

This paper contains 15 sections, 11 theorems, 32 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. Partitioning Into Subinstances
  4. Dynamic Programming
  5. Preliminaries: Notations, Quadtree, and Structure Properties
  6. Budget Multipath Problem
  7. First Algorithm: Dynamic Program with Budgets
  8. Construction
  9. Analysis
  10. Improved Algorithm and Proof of \ref{['thm:DP']}
  11. Construction
  12. Analysis
  13. Pseudocode for the First Algorithm
  14. Pseudocode for the Improved Algorithm
  15. Proof of \ref{['lem:RT']}

Key Result

Theorem 1

Let $d\geq 2$ be a fixed constant. For any $\varepsilon>0$, there is a randomized $(1+\varepsilon)$-approximation algorithm for the Euclidean $k$-TSP that runs in time The dependence on $\varepsilon$ in the running time is asymptotically optimal under the Gap-Exponential Time Hypothesis (Gap-ETH). The algorithm can be derandomized by increasing the running time by a factor $O(n^d)$.

Theorems & Definitions (28)

  • Theorem 1
  • Definition 2: well-rounded instance, arora1998polynomial
  • Theorem 3: partition theorem
  • Lemma 5: har2015net
  • Lemma 6
  • proof
  • proof : Proof of the partition theorem (\ref{['thm:partition']})
  • Theorem 7: dynamic programming theorem
  • Definition 8: grid, kisfaludi2022gap
  • Definition 9: fine multiset, adaptation from kisfaludi2022gap
  • ...and 18 more