[Paper Review] Approximation algorithms for TSP with neighborhoods in the plane

Adrian Dumitrescu, Joseph S. B. Mitchell|arXiv (Cornell University)|Jan 9, 2001

Computational Geometry and Mesh Generation25 references106 citations

TL;DR

This paper presents approximation algorithms for the Euclidean TSP with neighborhoods (TSPN), introducing a constant-factor approximation for connected neighborhoods with comparable diameters and a PTAS for disjoint or nearly disjoint unit disks. It also provides a linear-time O(1)-approximation for neighborhoods that are straight lines, significantly improving approximation ratios for several special cases of TSPN.

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.

Motivation & Objective

• To address the NP-hard TSPN problem by developing efficient approximation algorithms for geometric neighborhoods.
• To provide a constant-factor approximation for connected neighborhoods with comparable diameters.
• To present a polynomial-time approximation scheme (PTAS) for disjoint or nearly disjoint unit disk neighborhoods.
• To extend approximation techniques to geometric structures like infinite straight lines.
• To improve existing approximation ratios for special classes of neighborhoods previously studied.

Proposed method

• Designing a geometric clustering and tour construction approach tailored to neighborhoods with bounded diameter ratios.
• Applying a shifted hierarchical decomposition to handle unit disk neighborhoods and achieve a PTAS.
• Using a sweep-line and greedy selection strategy for neighborhoods defined by straight lines to achieve linear-time O(1)-approximation.
• Leveraging geometric properties such as convexity and proximity to bound approximation factors.
• Adapting existing TSP approximation techniques to the more general neighborhood setting via region covering and path merging.
• Integrating dynamic programming and local optimization within a hierarchical framework for unit disks.

Experimental results

Research questions

• RQ1 Can a constant-factor approximation be achieved for TSPN when neighborhoods are connected and have comparable diameters?
• RQ2 Is a PTAS possible for TSPN with disjoint or nearly disjoint unit disk neighborhoods?
• RQ3 Can a linear-time O(1)-approximation be designed for TSPN with line-segment or infinite line neighborhoods?
• RQ4 How do the approximation ratios for special neighborhood classes compare to prior work?
• RQ5 What geometric properties enable improved approximation guarantees in TSPN beyond the classical TSP?

Key findings

• A constant-factor approximation algorithm is achieved for connected neighborhoods with comparable diameters, improving on prior bounds.
• A polynomial-time approximation scheme (PTAS) is developed for disjoint or nearly disjoint unit disk neighborhoods.
• An O(1)-approximation algorithm runs in linear time for neighborhoods that are straight lines.
• The approximation ratio for unit disk neighborhoods is asymptotically optimal under standard complexity assumptions.
• The method yields improved approximation ratios for previously studied special cases, such as fat regions and convex sets.
• The approach generalizes effectively to various geometric neighborhood types through structural decomposition and geometric optimization.