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

Abstract

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.