arXiv subjects

Wagner da Rocha

Publications and source records attributed to Wagner da Rocha.

3 recordsLinked to original sources

arXiv2026

A Conditional Rank-Count Theory for the Combinatorial Discretizable Distance Geometry Problem

The Combinatorial Discretizable Distance Geometry Problem combines a finite binary lateration process with additional distance constraints. When predecessor sets are not consecutive, these pruning constraints interact in ways that make the symmetry-based counting methods available for molecular instances insufficient. We establish an exact, dimension-uniform counting theorem for seed-fixed feasible realizations under strict discretization and a generic feasible framework assumption. Our approach identifies partial reflections through seed-free connected components of the lateration skeleton. These reflections are encoded by binary masks, while a labeled constraint matrix detects combinations that preserve all pruning distances. The feasible branch choices split into constrained choices generated by compatible partial reflections and unconstrained choices outside the predecessor closure of the pruning endpoints. The resulting count is obtained from graph operations and rank computations over the binary field, without enumerating the lateration tree. Completeness follows from the characterization of generic realizations of the relevant joined graphs by partial reflections. The theorem applies in every Euclidean dimension, including dimension one.

math.CO↗

arXiv2026

A Rank-Count Theory for the Combinatorial Discretizable Distance Geometry Problem

The Distance Geometry Problem (DGP) asks for a geometric realization of a weighted graph with \(n\) vertices in \(\mathbb{R}^K\) such that Euclidean distances between vertices match the given edge weights. When a vertex order where every non-seed vertex has \(K\) predecessors inducing a clique is part of the input, the search space can be discretized via \(K\)-lateration, branching into a binary tree. In this subclass, known as the Combinatorial Discretizable Distance Geometry Problem (Combinatorial DDGP), the goal is to determine the number of realizations satisfying all distance constraints. While the number of realizations is almost always \(2^{n-K}\) when no additional distance constraints are present, a topological solution count in the presence of additional constraints has remained elusive. We develop an algebraic rank-count theory for the feasible binary branch codes, proving that under mirror-separated parameters they form an affine space over \(\mathbb{F}_2\) whenever a viable reference solution exists.

math.MG↗

arXiv2025

A hybrid combinatorial-continuous strategy for solving molecular distance geometry problems

The Molecular Distance Geometry Problem (MDGP) is essential in structural biology, as it seeks to determine three-dimensional protein structures from partial interatomic distances. Its discretizable subclass (DMDGP) admits an exact combinatorial formulation that enables efficient exploration of the search space. However, in practical settings such as Nuclear Magnetic Resonance (NMR) spectroscopy, distances are available only within uncertainty bounds, leading to the interval variant (\emph{i}DMDGP). We propose a hybrid combinatorial--continuous framework for solving the \emph{i}DMDGP. The method combines an enumeration process derived from the DMDGP with a continuous refinement stage that minimizes a nonconvex stress function that penalizes deviations from admissible distance intervals. This integration supports a systematic exploration guided by discrete structure and local optimization. The formulation incorporates torsion-angle intervals and chirality constraints through a refined atom ordering that preserves protein-backbone geometry. Numerical experiments show that the approach efficiently reconstructs geometrically valid conformations even under wide distance bounds, whereas most existing studies assume narrow ones.

math.OC↗