Abstract
Given a weighted undirected graph \(G = (V,E,d)\) with \(d : E \rightarrow {\mathbb{Q}}_{+}\) and a positive integer K, the distance geometry problem (DGP) asks to find an embedding \(x : V \rightarrow {\mathbb{R}}^{K}\) of G such that for each edge \(\{i,j\}\) we have \(\|{x}_{i} - {x}_{j}\| = {d}_{ij}\). Saxe proved in 1979 that the DGP is NP-complete with K = 1 and doubted the applicability of the Turing machine model to the case with K > 1, because the certificates for YES instances might involve real numbers. This chapter is an account of an unfortunately failed attempt to prove that the DGP is in NP for K = 2. We hope that our failure will motivate further work on the question.
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Beeker, N., Gaubert, S., Glusa, C., Liberti, L. (2013). Is the Distance Geometry Problem in NP?. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds) Distance Geometry. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5128-0_5
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