### 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 ℝ^{2}. 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 O(logn)-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 O(log ^{d+1} n)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best O(n ^{ε} )-ratio for MMN in d dimensions [ESA'11]. En route, we show that an existing O(logn)-approximation algorithm for 2D-RSA generalizes to higher dimensions.

Original language | English (US) |
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Title of host publication | Algorithms and Computation - 24th International Symposium, ISAAC 2013, Proceedings |

Pages | 722-732 |

Number of pages | 11 |

DOIs | |

State | Published - Dec 1 2013 |

Event | 24th International Symposium on Algorithms and Computation, ISAAC 2013 - Hong Kong, China Duration: Dec 16 2013 → Dec 18 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8283 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 24th International Symposium on Algorithms and Computation, ISAAC 2013 |
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Country | China |

City | Hong Kong |

Period | 12/16/13 → 12/18/13 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Algorithms and Computation - 24th International Symposium, ISAAC 2013, Proceedings*(pp. 722-732). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8283 LNCS). https://doi.org/10.1007/978-3-642-45030-3_67