Minimum-size DAG or network realization

Determine the computational complexity of finding a directed acyclic graph or phylogenetic network that realizes a given relation R with the minimum number of vertices.

Background

The paper constructs a canonical DAG G_R and a canonical network N_R that realize every realizable LCA-constraint relation R in polynomial time. The authors show, however, that the canonical DAG need not have the fewest vertices among all realizing DAGs. They therefore identify minimum-size realization as an unresolved optimization problem.

References

Determining the computational complexity of finding a DAG or network realizing $R$ with the minimum number of vertices thus remains an open problem.

— Inferring DAGs and Phylogenetic Networks from Least Common Ancestors  (2511.07965 - Lindeberg et al., 11 Nov 2025) in Section 8, Discussion and Outlook, paragraph “Underlying Optimization Problems”