Is the Distance Geometry Problem in NP?

Nathanael Beeker (), Stéphane Gaubert (), Christian Glusa () and Leo Liberti ()
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Nathanael Beeker: CMAP, École Polytechnique
Stéphane Gaubert: INRIA Rocquencourt
Christian Glusa: CMAP, École Polytechnique
Leo Liberti: LIX, École Polytechnique

Chapter Chapter 5 in Distance Geometry, 2013, pp 85-93 from Springer

Abstract: 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.

Keywords: Distance Geometry Problem (DGP); Molecular Distance Geometry Problem (MDGP); Wireless Sensor Network Localization (WSNL); Meaningful Strings; Polynomial System Representation (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-5128-0_5

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