Randomized trial shows NP-completeness of geometric problems with varied distance metrics, suggesting important implications for computational complexity.
We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NP-complete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NP-hard if the distance measure is the (unmodified) Euclidean metric. However, for reasons we discuss, there is some question as to whether these problems, or even the well-solved MINIMUM SPANNING TREE problem, are in NP when the distance measure is the Euclidean metric.
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Garey et al. (1976) studied this question.
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