### Abstract

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(n^{5}) is known [28], settling the exact approx-imability 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.

Original language | English |
---|---|

Title of host publication | ACM-SIAM Symposium on Discrete Algorithms |

Publisher | ACM Publishing |

Pages | 1089-1105 |

Number of pages | 17 |

ISBN (Electronic) | 978-1-61197-548-2 |

DOIs | |

Publication status | Published - 2019 |

Externally published | Yes |

### Fingerprint

### Cite this

*ACM-SIAM Symposium on Discrete Algorithms*(pp. 1089-1105). ACM Publishing. https://doi.org/10.1137/1.9781611975482.67

}

*ACM-SIAM Symposium on Discrete Algorithms.*ACM Publishing, pp. 1089-1105. https://doi.org/10.1137/1.9781611975482.67

**A PTAS for Euclidean TSP with hyperplane neighborhoods.** / Antoniadis, Antonios; Fleszar, Krzysztof; Hoeksma, Ruben; Schewior, Kevin.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

TY - GEN

T1 - A PTAS for Euclidean TSP with hyperplane neighborhoods

AU - Antoniadis, Antonios

AU - Fleszar, Krzysztof

AU - Hoeksma, Ruben

AU - Schewior, Kevin

PY - 2019

Y1 - 2019

N2 - 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 approx-imability 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.

AB - 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 approx-imability 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.

U2 - 10.1137/1.9781611975482.67

DO - 10.1137/1.9781611975482.67

M3 - Conference contribution

AN - SCOPUS:85066953198

SP - 1089

EP - 1105

BT - ACM-SIAM Symposium on Discrete Algorithms

PB - ACM Publishing

ER -