### Abstract

We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in (Formula presented.). The goal is to find a minimum-length rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis-parallel line segments whose total length equals the pair’s Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the well-known rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NP-hard. No approximation algorithms are known for general GMMN. We obtain an (Formula presented.)-approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple (Formula presented.)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best (Formula presented.)-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing (Formula presented.)-approximation algorithm for 2D-RSA generalizes to higher dimensions.

Language | English (US) |
---|---|

Pages | 1-21 |

Number of pages | 21 |

Journal | Algorithmica |

DOIs | |

State | Accepted/In press - Mar 2 2017 |

### Fingerprint

### Keywords

- Approximation algorithms
- Computational geometry
- Minimum Manhattan Network

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica*, 1-21. DOI: 10.1007/s00453-017-0298-0

**Approximating the Generalized Minimum Manhattan Network Problem.** / Das, Aparna; Fleszar, Krzysztof; Kobourov, Stephen; Spoerhase, Joachim; Veeramoni, Sankar; Wolff, Alexander.

Research output: Research - peer-review › Article

*Algorithmica*, pp. 1-21. DOI: 10.1007/s00453-017-0298-0

}

TY - JOUR

T1 - Approximating the Generalized Minimum Manhattan Network Problem

AU - Das,Aparna

AU - Fleszar,Krzysztof

AU - Kobourov,Stephen

AU - Spoerhase,Joachim

AU - Veeramoni,Sankar

AU - Wolff,Alexander

PY - 2017/3/2

Y1 - 2017/3/2

N2 - We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in (Formula presented.). The goal is to find a minimum-length rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis-parallel line segments whose total length equals the pair’s Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the well-known rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NP-hard. No approximation algorithms are known for general GMMN. We obtain an (Formula presented.)-approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple (Formula presented.)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best (Formula presented.)-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing (Formula presented.)-approximation algorithm for 2D-RSA generalizes to higher dimensions.

AB - We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in (Formula presented.). The goal is to find a minimum-length rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis-parallel line segments whose total length equals the pair’s Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the well-known rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NP-hard. No approximation algorithms are known for general GMMN. We obtain an (Formula presented.)-approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple (Formula presented.)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best (Formula presented.)-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing (Formula presented.)-approximation algorithm for 2D-RSA generalizes to higher dimensions.

KW - Approximation algorithms

KW - Computational geometry

KW - Minimum Manhattan Network

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U2 - 10.1007/s00453-017-0298-0

DO - 10.1007/s00453-017-0298-0

M3 - Article

SP - 1

EP - 21

JO - Algorithmica

T2 - Algorithmica

JF - Algorithmica

SN - 0178-4617

ER -