In this paper we investigate the complexity of embedding edge-weighted graphs into Euclidean spaces: Given an (incomplete) edge-weighted graph, G, can the vertices of G be mapped to points in Euclidean k-space in such a way that any two vertices connected by an edge are mapped to points whose distance is equal to the weight of the edge? We prove that the preceding problem is NP-Hard (by reduction from 3-Satisfiability), even when k=1 and the edge weights are restricted to take on the values 1 and 2. Related results are shown for the problem of testing the uniqueness of a known embedding and for variations involving inexact edge weights.
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