TY - GEN

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:
This work was supported by the ESF EuroGIGA project GraDR (DFG grant Wo 758/5-1).

PY - 2013

Y1 - 2013

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(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.

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(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.

UR - http://www.scopus.com/inward/record.url?scp=84893421914&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893421914&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-45030-3_67

DO - 10.1007/978-3-642-45030-3_67

M3 - Conference contribution

AN - SCOPUS:84893421914

SN - 9783642450297

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 722

EP - 732

BT - Algorithms and Computation - 24th International Symposium, ISAAC 2013, Proceedings

T2 - 24th International Symposium on Algorithms and Computation, ISAAC 2013

Y2 - 16 December 2013 through 18 December 2013

ER -