The Traveling Salesman Problem Under Squared Euclidean Distances

TL;DR

This paper introduces approximation algorithms for the Traveling Salesman Problem under squared Euclidean distances, providing new bounds for specific dimensions and exponents, and explores the revisiting variant's complexity.

Contribution

The paper develops a 5-approximation for TSP(2,2), generalizes it to a broader class, and presents a PTAS for Rev-TSP(2,α), also establishing APX-hardness for higher dimensions.

Findings

5-approximation for TSP(2,2)

Generalized approximation factor for TSP(2,α)

Polynomial-time approximation scheme for Rev-TSP(2,α)

Abstract

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.