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The Minimal Manhattan Network Problem in Three Dimensions

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WALCOM: Algorithms and Computation (WALCOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5431))

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Abstract

For the Minimal Manhattan Network Problem in three dimensions (MMN3D), one is given a set of points in space, and an admissible solution is an axis-parallel network that connects every pair of points by a shortest path under L 1-norm (Manhattan metric). The goal is to minimize the overall length of the network.

Here, we show that MMN3D is \(\cal NP\)- and \(\cal APX\)-hard, with a lower bound on the approximability of 1 + 2·10− 5.

This lower bound applies to MMN2-3D already, a sub-problem in between the two and three dimensional case. For MMN2-3D, we also develop a 3-approximation algorithm which is the first algorithm for the Minimal Manhattan Network Problem in three dimensions at all.

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Muñoz, X., Seibert, S., Unger, W. (2009). The Minimal Manhattan Network Problem in Three Dimensions. In: Das, S., Uehara, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2009. Lecture Notes in Computer Science, vol 5431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00202-1_32

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  • DOI: https://doi.org/10.1007/978-3-642-00202-1_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00201-4

  • Online ISBN: 978-3-642-00202-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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