Characterization and construction of maximal admissible hierarchical clustering methods

Determine which additional hierarchical clustering methods satisfy scale invariance, partition richness, partition consistency, and permutation invariance, and construct explicit maximal methods within the refinement-ordered class of admissible hierarchical clustering methods.

Background

The paper establishes that the hierarchical analogs of Kleinberg’s axioms are jointly satisfiable and that the resulting class of admissible methods is uncountable and structurally diverse. It also proves that maximal admissible methods exist and that there are uncountably many pairwise-incompatible maximal elements, but it does not identify explicit examples of such maximal methods or classify the full admissible class.

The open problem therefore concerns both the broader membership question—determining which hierarchical clustering procedures beyond those constructed in the paper are admissible—and the constructive order-theoretic question of giving explicit descriptions or algorithms for maximal admissible methods.

References

This study raises several open questions. A first one is to determine which other hierarchical clustering methods are admissible, and, in particular, to identify explicit maximal admissible methods.

— Hierarchical Clustering Can Jointly Satisfy Richness, Consistency, and Scale Invariance  (2609.11173 - Kuroda et al., 10 Sep 2026) in Section 6, Conclusion

This study raises several open questions. A first one is to determine which other hierarchical clustering methods are admissible, and, in particular, to identify explicit maximal admissible methods. Another one is to understand which additional axioms meaningfully reduce the large admissible class.

— Hierarchical Clustering Can Jointly Satisfy Richness, Consistency, and Scale Invariance  (2609.11173 - Kuroda et al., 10 Sep 2026) in Section 6, Conclusion