# Approximation algorithms for TSP with neighborhoods in the plane

## TL;DR

This paper introduces new approximation algorithms for the Euclidean TSP with neighborhoods, including a constant-factor algorithm for connected regions, a PTAS for disjoint unit disks, and an efficient line-based solution.

## Contribution

It provides the first constant-factor approximation for connected neighborhoods and a PTAS for disjoint unit disks in TSPN, advancing the understanding of approximation in geometric TSP variants.

## Key findings

- Constant-factor approximation for connected neighborhoods
- PTAS for disjoint unit disk neighborhoods
- Linear-time O(1)-approximation for line neighborhoods

## Abstract

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NP-hard. In this paper, we present new approximation results for the TSPN, including (1) a constant-factor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearly-unit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a linear-time O(1)-approximation algorithm for the case of neighborhoods that are (infinite) straight lines.