On the polynomiality of finding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e37" altimg="si4.svg"><mml:msup><mml:mrow/><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msup></mml:math>DMDGP re-orders

Souza, Michael; Carvalho, Luiz Mariano; Liberti, Leo

doi:10.1016/j.dam.2019.07.021

Cited by 11 publications

(3 citation statements)

References 31 publications

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“…The DGP is naturally cast as a search in continuous space. Depending on the graph structure, however, combinatorial search algorithms can be defined, notably via the identification of appropriate vertex orders [5,11,14]. Although DGP is NP-hard [30], these combinatorial approaches allowed to show that it is Fixed Parameter Tractable (FPT) on certain graph structures, as those arising in protein conformation [20].…”

Section: Assumptionmentioning

confidence: 99%

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

2021

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The fundamental inverse problem in distance geometry is the one of finding positions from inter-point distances. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem (DGP) whose search space can be discretized and represented by a binary tree, which can be explored by a Branch-and-Prune (BP) algorithm. It turns out that this combinatorial search space possesses many interesting symmetry properties that were studied in the last decade. In this paper, we present a new algorithm for this subclass of the DGP, which exploits DMDGP symmetries more effectively than its predecessors. Computational results show that the speedup, with respect to the classic BP algorithm, is considerable for sparse DMDGP instances related to protein conformation.

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Section: Assumptionmentioning

confidence: 99%

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

2021

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“…The BP algorithm was first proposed for CTOP by Lavor et al [9], and in [3] answer set programming was shown to improve the performance of the BP algorithm for CTOP. Furthermore, DMDGP vertex orders with repeated vertices, called re-orders, are considered [10], their computational complexity is analyzed [1] and further studied in detail [11].…”

Section: Introductionmentioning

confidence: 99%

Constraint programming approaches for the discretizable molecular distance geometry problem

MacNeil

Bodur

2021

Networks

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The Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so‐called discretization assumptions allow us to discretize the search space of DGP instances. In this paper, we focus on a key subclass of DGP, namely the Discretizable Molecular DGP, and study its associated graph vertex ordering problem, the Contiguous Trilateration Ordering Problem (CTOP), which helps solve DGP. We propose the first constraint programming formulations for CTOP, as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints, and valid inequalities. Our computational results on random and pseudo‐protein instances indicate that our formulations outperform the state‐of‐the‐art integer programming formulations.

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“…where f is a function f : We may furnish the set V of vertices with an ordering V = {v , ..., vn} [9,15,23,25] so that the MDGP can be solved iteratively using a combinatorial method, namely the Branch-and-Prune (BP) method [8,28]. In this situation, the MDGP is called the Discretizable Molecular Distance Geometry Problem (DMDGP) [19,20], which can be stated as follows, where we use x i instead of xv i and d i,j in place of dv i ,v j : (DMDGP) Given a simple undirected graph G = (V , E, d) in which the vertex set V is ordered as V = {v , ..., vn}, whose edges are weighted by d : E → ( , ∞), subject to the following three constraints:…”

Section: Introductionmentioning

confidence: 99%

NMR Protein Structure Calculation and Sphere Intersections

Alves

Souza

José

2020

Computational and Mathematical Biophysics

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Nuclear Magnetic Resonance (NMR) experiments can be used to calculate 3D protein structures and geometric properties of protein molecules allow us to solve the problem iteratively using a combinatorial method, called Branch-and-Prune (BP). The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i − 3, i − 2, i − 1, with radii given by the atomic distances di−3,i, di−2,i, di−1,i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di−3,i should be represented as interval distances [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}], where {\underline{d}_{i - 3,i}} \le {d_{i - 3,i}} \le {\bar d_{i - 3,i}}. In the literature, an extension of the BP algorithm was proposed to deal with interval distances, where the idea is to sample values from [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}]. We present a new method, based on conformal geometric algebra, to reduce the size of [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}], before the sampling process. We also compare it with another approach proposed in the literature.

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On the polynomiality of finding KDMDGP re-orders

Abstract: In [6], the complexity of finding K DMDGP reorders was stated to be NP-complete by inclusion, which fails to provide a complete picture. In this paper we show that this problem is indeed NP-complete for K = 1, but it is in P for each fixed K ≥ 2.

Cited by 11 publications

References 31 publications

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

Constraint programming approaches for the discretizable molecular distance geometry problem

NMR Protein Structure Calculation and Sphere Intersections

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