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

N1 - Funding Information:
A preliminary version of this paper appeared in Proc. 24th International Symposium on Algorithms and Complexity (ISAAC’13), volume 8283 of Lect. Notes Comput. Sci., pp. 722–732. This work was supported by the ESF EuroGIGA project GraDR (DFG Grant Wo 758/5-1).
Publisher Copyright:
© 2017, Springer Science+Business Media New York.

PY - 2018/4/1

Y1 - 2018/4/1

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 R2. 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(log n) -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 + 1n) -approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best O(nε) -ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing O(log n) -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 R2. 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(log n) -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 + 1n) -approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best O(nε) -ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing O(log n) -approximation algorithm for 2D-RSA generalizes to higher dimensions.

KW - Approximation algorithms

KW - Computational geometry

KW - Minimum Manhattan Network

UR - http://www.scopus.com/inward/record.url?scp=85014075563&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014075563&partnerID=8YFLogxK

U2 - 10.1007/s00453-017-0298-0

DO - 10.1007/s00453-017-0298-0

M3 - Article

AN - SCOPUS:85014075563

VL - 80

SP - 1170

EP - 1190

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 4

ER -