Computational complexity of dominated ultrametric feasibility

Determine the precise computational complexity of deciding whether a dominating ultrametric satisfies a given system of pairwise upper-bound constraints, and identify tractable structural subclasses of this feasibility problem.

Background

The paper formulates individually fair hierarchical clustering as the search for an ultrametric that dominates a given metric while satisfying local upper-bound constraints on designated nearest-neighbor pairs. For a fixed dendrogram, feasibility can be checked in polynomial time by verifying the induced inequalities. However, the unrestricted problem requires searching over all dendrogram structures, whose merge heights are globally coupled by the ultrametric condition.

The paper explicitly notes that local consistency of the constraints does not ensure global realizability, as illustrated by the separation between local fairness thresholds and the global distortion required by ultrametric embeddings. Consequently, the precise complexity of deciding feasibility over all ultrametrics, as well as the identification of efficiently solvable structural subclasses, is left unresolved.