Approximating the Generalized Minimum Manhattan Network Problem

  • Aparna Das
  • Krzysztof Fleszar
  • Stephen Kobourov
  • Joachim Spoerhase
  • Sankar Veeramoni
  • Alexander Wolff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


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.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Aparna Das
    • 1
  • Krzysztof Fleszar
    • 2
  • Stephen Kobourov
    • 1
  • Joachim Spoerhase
    • 2
  • Sankar Veeramoni
    • 1
  • Alexander Wolff
    • 2
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonU.S.A.
  2. 2.Lehrstuhl I, Institut für InformatikUniversität WürzburgGermany

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