On the Complexity of the Ordered Covering Problem in Distance Geometry
Michael Souza, Júlio Araújo, John Kesley Costa, Carlile Lavor
TL;DR
This paper proves that the Ordered Covering Problem (OCP), arising from the edge-ordering aspect of the SBBU algorithm in the Discretizable Molecular Distance Geometry Problem (DMDGP), is NP-complete via a polynomial-time reduction from 3-Partition. The construction uses an exponential cost structure and a tight budget to enforce a one-to-one correspondence between feasible OCP solutions and valid 3-partitions, with opening and assignment edges forcing a triplet-based coverage. The result explains why greedy heuristics and pruning-based strategies perform well in practice, as no polynomial-time algorithm can guarantee optimal edge ordering unless P = NP. It also situates OCP among distance-geometry complexity results and motivates further work on approximation guarantees and parameterized strategies for practical instances.
Abstract
The Ordered Covering Problem (OCP) arises in the context of the Discretizable Molecular Distance Geometry Problem (DMDGP), where the ordering of pruning edges significantly impacts the performance of the SBBU algorithm for protein structure determination. In recent work, Souza et al. (2023) formalized OCP as a hypergraph covering problem with ordered, exponential costs, and proposed a greedy heuristic that outperforms the original SBBU ordering by orders of magnitude. However, the computational complexity of finding optimal solutions remained open. In this paper, we prove that OCP is NP-complete through a polynomial-time reduction from the strongly NP-complete 3-Partition problem. Our reduction constructs a tight budget that forces optimal solutions to correspond exactly to valid 3-partitions. This result establishes a computational barrier for optimal edge ordering and provides theoretical justification for the heuristic approaches currently used in practice.