Approximation algorithms for TSP with neighborhoods in the plane

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.

• The paper presents constant-factor approximation algorithms, a PTAS for disjoint unit disks, and improved bounds for various classes of TSP with Neighborhoods (TSPN) in the plane.
• Methodological innovations include extended m-guillotine methods and geometric packing arguments to derive near-optimal solutions and establish approximation bounds.
• The research offers practical algorithms for geometric TSPN instances and lays a foundation for addressing similar problems in higher dimensions or with more complex neighborhood structures.

Approximation Algorithms for Traveling Salesman Problem with Neighborhoods in the Plane

The paper authored by Adrian Dumitrescu and Joseph S. B. Mitchell addresses the Traveling Salesman Problem with Neighborhoods (TSPN), a natural extension of the classical Euclidean Traveling Salesman Problem (TSP). In TSPN, the tour must visit a collection of specified regions, or "neighborhoods," in the plane, rather than discrete points. The problem is known to be NP-hard, similar to its classical counterpart. The study presents novel approximation algorithms for various classes of this problem, providing insights into its complexity and computability.

Key Contributions and Results

1. Constant-Factor Approximation for Connected Neighborhoods with Comparable Diameters: The paper introduces a constant-factor approximation algorithm applicable when the neighborhoods have connected regions with similar diameters. This result addresses an open problem in previous research on approximating TSPN for certain types of segments.
2. Polynomial Time Approximation Scheme (PTAS) for Disjoint Unit Disks: For the specific case of disjoint unit disk neighborhoods, or nearly disjoint disks of nearly the same size, the authors propose a PTAS. This is a significant result as it provides an efficient approximation scheme, particularly leveraging a refined area-based charging method in the analysis.
3. Improved Bounds for Special Classes: The paper extends previous results by offering improved approximation ratios for previously studied special cases, including parallel segments and translates of convex or connected regions.
4. Linear-Time Approximation for Infinite Lines: A unique linear-time algorithm is provided that offers an O(1) approximation for the case where neighborhoods are infinite straight lines. The approach demonstrates a practical solution for a case with theoretical challenges.

Methodological Innovations

• m-Guillotine Methods and Structural Results: By extending the concept of m-guillotine subdivisions, the authors develop novel structural theorems that facilitate the derivation of near-optimal solutions for neighborhoods consisting of equal-sized disks. These methods allow breaking down the problem into manageable subproblems that can be solved recursively.
• Packing Arguments: For disjoint neighborhoods, the authors utilize geometric packing arguments that relate the area covered by optimal solutions to the number of neighborhoods. This approach is pivotal in establishing the approximation bounds provided in the paper.

Theoretical and Practical Implications

The research provides foundational tools for addressing complex geometric instances of the TSPN. By focusing on various specialized cases and providing robust approximation algorithms, this work significantly enhances the potential to solve practical instances of TSPN in logistics, robotics, and other fields requiring efficient routing in geometric environments.

The PTAS for disjoint and nearly equal disks stands as a notable theoretical contribution. It highlights a pathway toward resolving similar problems with neighborhood structures in higher dimensions or more general forms, where the geometric nature of the problem can be judiciously exploited.

Future Directions

The paper concludes by outlining several open questions, such as developing constant-factor approximation algorithms for more general connected regions or extending the results to higher dimensions. Moreover, exploring PTAS approaches for broader classes of regions remains an open field. These areas invite further investigation, potentially leading to improvements in computational efficiency and broader applicability of the proposed methods for generalized geometric optimization problems.

Overall, this research delineates a comprehensive approach for approximating TSPN in the plane, providing both theoretical advancements and applicable algorithms, thus enriching the landscape of combinatorial optimization in Euclidean spaces.