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
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