Free access
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

A PTAS for Euclidean TSP with Hyperplane Neighborhoods


In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given a collection of geometric regions in some space. The goal is to output a tour of minimum length that visits at least one point in each region. Even in the Euclidean plane, TSPN is known to be APX-hard [20], which gives rise to studying more tractable special cases of the problem. In this paper, we focus on the fundamental special case of regions that are hyperplanes in the d-dimensional Euclidean space. This case contrasts the much-better understood case of so-called fat regions [16, 34].
While for d = 2 an exact algorithm with running time O(n5) is known [28], settling the exact approximability of the problem for d = 3 has been repeatedly posed as an open question [23, 24, 34, 40]. To date, only an approximation algorithm with guarantee exponential in d is known [24], and NP-hardness remains open.
For arbitrary fixed d, we develop a Polynomial Time Approximation Scheme (PTAS) that works for both the tour and path version of the problem. Our algorithm is based on approximating the convex hull of the optimal tour by a convex polytope of bounded complexity. Such polytopes are represented as solutions of a sophisticated LP formulation, which we combine with the enumeration of crucial properties of the tour. As the approximation guarantee approaches 1, our scheme adjusts the complexity of the considered polytopes accordingly.
In the analysis of our approximation scheme, we show that our search space includes a sufficiently good approximation of the optimum. To do so, we develop a novel and general sparsification technique to transform an arbitrary convex polytope into one with a constant number of vertices and, in turn, into one of bounded complexity in the above sense. Hereby, we maintain important properties of the polytope.

Formats available

You can view the full content in the following formats:

Information & Authors


Published In

cover image Proceedings
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 1089 - 1105
Editor: Timothy M. Chan, University of Illinois at Urbana-Champaign, USA
ISBN (Online): 978-1-61197-548-2


Published online: 2 January 2019




The first author is supported by DFG grant AN 1262/1-1. The second and fourth author were supported by CONICYT Grant PII 20150140. The fourth author is also supported within the DAAD PRIME program.

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options


View PDF

Get Access







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account