Minimum Manhattan Network is NP-Complete

Authors:
Francis Y. L. Chin;Zeyu Guo;He Sun
Affiliations:
The University of Hong Kong, Department of Computer Science, Pokfulam, Hong Kong;Fudan University, School of Computer Science, Shanghai, China;Fudan University, School of Computer Science, Shanghai, China
Venue:
Discrete & Computational Geometry - Special Issue: 25th Annual Symposium on Computational Geometry; Guest Editor: John Hershberger
Year:
2011

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Abstract

Given a set T of n points in ℝ2, a Manhattan network on T is a graph G with the property that for each pair of points in T, G contains a rectilinear path between them of length equal to their distance in the L 1-metric. The minimum Manhattan network problem is to find a Manhattan network of minimum length, i.e., minimizing the total length of the line segments in the network. In this paper, we prove that the decision version of the MMN problem is strongly NP-complete, using a reduction from the well-known 3-SAT problem, which requires a number of gadgets. The gadgets have similar structures, but play different roles in simulating a 3-CNF formula.