The K-discretization and K-incident graphs for discretizable Distance Geometry

Abud, Germano; Alencar, Jorge; Liberti, Leo; Mucherino, Antonio

doi:10.1007/s11590-018-1294-2

Cited by 3 publications

(3 citation statements)

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“…α = [1. 1,2]. For more complicated graphs it is possible to have situations where both endpoints of the interval yield realizations of deficient ranks.…”

Section: What We Mean By "Counting"mentioning

confidence: 99%

“…The n × n symmetric zero-diagonal matrix having x i − x j 2 2 as its (i, j)-th entry is a squared Euclidean Distance Matrix (EDM). It turns out that G = − 1 2 JDJ (where J = I n − 1 n 11 is the centering matrix and 1 is the all-one vector) is the Gram matrix of the realization x, i.e. xx = G [33,9].…”

Section: Introductionmentioning

confidence: 99%

“…The techniques used for the structure determination of the discretization and pruning groups are specific to the DMDGP. No easy extension to the DDGP was found so far using those techniques (see [1] for an attempt). In this paper, we propose a new theoretical analysis of the number of solutions of the DDGP.…”

Section: Introductionmentioning

confidence: 99%

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An impossible utopia in distance geometry

Abud¹,

Alencar²,

Liberti³

et al. 2021

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The Distance Geometry Problem asks for a realization of a given weighted graph in R K . Two variants of this problem, both originating from protein conformation, are based on a given vertex order (which abstracts the protein backbone). Both variants involve an element of discrete decision in the realization of the next vertex in the order using K preceding (already realized) vertices. The difference between these variants is that one requires the K preceding vertices to be contiguous. The presence of this constraint allows one to prove, via a combinatorial counting of the number of solutions, that the realization algorithm is fixed-parameter tractable. Its absence, on the other hand, makes it possible to efficiently construct the vertex order directly from the graph. Deriving a combinatorial counting method without using the contiguity requirement would therefore be desirable. In this paper we prove that, unfortunately, such a counting method cannot be devised in general.

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“…α = [1. 1,2]. For more complicated graphs it is possible to have situations where both endpoints of the interval yield realizations of deficient ranks.…”

Section: What We Mean By "Counting"mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%