On the Complexity of the Ordered Covering Problem in Distance Geometry

TL;DR

This paper proves that the Ordered Covering Problem in Distance Geometry is NP-complete, explaining the computational difficulty of finding optimal orderings and justifying heuristic methods used in protein structure determination.

Contribution

It establishes the NP-completeness of OCP via a polynomial reduction from 3-Partition, providing a theoretical foundation for heuristic approaches.

Findings

OCP is NP-complete.

Optimal solutions correspond to valid 3-partitions.

Heuristic methods are justified by computational complexity.

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.