The paper investigates whether the impossibility results for flat clustering—specifically Kleinberg’s axioms of scale invariance, richness, and consistency—extend to hierarchical clustering. It demonstrates that, unlike the flat case, there exist uncountably many hierarchical clustering methods that satisfy all three axioms, termed admissible methods. The authors construct several such methods, explore a refinement partial order among them, and show that while the set of admissible methods is diverse, every method shares a common backbone of well‑separated clusters.
Consider the following variation on the Hierarchical Clustering problem: Usually, while building a hierarchical clustering, one recursively partitions the data until each cluster becomes a singleton. We relax the halting condition of the recursive process to stop whenever the remaining cluster is a graph belonging to a class $\mathcal{F}$.
arXiv:2607. 13217v1 Announce Type: cross Abstract: Consider the following variation on the Hierarchical Clustering problem: Usually, while building a hierarchical clustering, one recursively partitions the data until each cluster becomes a singleton.
By Micha{\l} Szyfelbein, Dariusz Dereniowski
arXiv:2604. 23628v2 Announce Type: replace-cross Abstract: Hierarchical clustering is a fundamental task in data analysis, but classical methods have long lacked a principled objective function.
By Ryuki Tsukuba, Kazutoshi Ando
arXiv:2607. 10305v1 Announce Type: cross Abstract: Byzantine-robust aggregation rules such as multi-Krum assume a central coordinator, and decentralising them is obstructed by the rules themselves: they are globally coupled, non-associative, and discontinuous, so an ulpscale perturbation can flip the selected subset, moving the output by a non-vanishing amount.
By Ryan Gillespie
The paper investigates hierarchical clustering under an individual fairness constraint that limits relative distortion within local k‑nearest neighborhoods. It formulates this as a feasibility problem over dominated ultrametrics, characterizes the minimal multiplicative slack needed, identifies a sharp local threshold, proves stability under bounded perturbations, establishes monotonicity in k, and demonstrates a Θ(log n) separation between local and global realizability. Experiments on synthetic and real‑world datasets corroborate the theoretical findings.
By Binita Maity, Shrutimoy Das
The paper introduces the Universal Clustering Problem (UCP), a framework that captures the optimisation core common to many clustering methods by maximizing a polynomial‑time computable partition utility over a finite metric space. It proves UCP is NP‑hard through reductions from graph colouring and exact cover by 3‑sets, showing that popular algorithms such as k‑means, GMMs, DBSCAN, spectral clustering, and affinity propagation inherit this intractability. The authors argue that this unified hardness explains typical failure modes—like local optima and greedy merge traps—and suggest moving toward stability‑aware objectives and interaction‑driven formulations with explicit guarantees.
By Angshul Majumdar
arXiv:2606. 18972v1 Announce Type: cross Abstract: Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem.
By Connor Simpson, Ricardo J. G. B. Campello
Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem. Existing approaches focus on finding a single optimal solution.
arXiv:2608. 04014v1 Announce Type: cross Abstract: The subdominant (minmax) ultrametric is a canonical tree-structured summary of a dissimilarity matrix, arising equivalently as the ultrametric induced by single-linkage clustering.
By Alokendu Mazumder, Arnab Roy, Punit Rathore
arXiv:2608.29097v1 Announce Type: cross
Abstract: This paper studies the problem of proportionally fair clustering, where the goal is to select $k$ ``centers'' from a metric space that fairly represe...
By Benjamin Cookson, Eva Deltl, Yeeseok Oh
arXiv:2609.06127v1 Announce Type: cross
Abstract: Retrieval-augmented generation is usually formulated as a statistical information-retrieval problem. Graph-based variants add relational structure, b...
By Xingting Wang, Min Wu