Approximating the Generalized Minimum Manhattan Network Problem

Aparna Das, Krzysztof Fleszar, Stephen Kobourov, Joachim Spoerhase, Sankar Veeramoni, Alexander Wolff

Research output: Research - peer-reviewArticle

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.

LanguageEnglish (US)
Pages1-21
Number of pages21
JournalAlgorithmica
DOIs
StateAccepted/In press - Mar 2 2017

Fingerprint

Approximation algorithms
Approximation Algorithms
Computational complexity
Steiner's problem
Higher Dimensions
Path
Generalise
Generalization
Line segment
Corollary
NP-complete problem
Arbitrary

Keywords

  • Approximation algorithms
  • Computational geometry
  • Minimum Manhattan Network

ASJC Scopus subject areas

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

Cite this

Das, A., Fleszar, K., Kobourov, S., Spoerhase, J., Veeramoni, S., & Wolff, A. (2017). Approximating the Generalized Minimum Manhattan Network Problem. 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.

In: Algorithmica, 02.03.2017, p. 1-21.

Research output: Research - peer-reviewArticle

Das, A, Fleszar, K, Kobourov, S, Spoerhase, J, Veeramoni, S & Wolff, A 2017, 'Approximating the Generalized Minimum Manhattan Network Problem' Algorithmica, pp. 1-21. DOI: 10.1007/s00453-017-0298-0
Das A, Fleszar K, Kobourov S, Spoerhase J, Veeramoni S, Wolff A. Approximating the Generalized Minimum Manhattan Network Problem. Algorithmica. 2017 Mar 2;1-21. Available from, DOI: 10.1007/s00453-017-0298-0
Das, Aparna ; Fleszar, Krzysztof ; Kobourov, Stephen ; Spoerhase, Joachim ; Veeramoni, Sankar ; Wolff, Alexander. / Approximating the Generalized Minimum Manhattan Network Problem. In: Algorithmica. 2017 ; pp. 1-21
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