Universal NP-Hardness of Clustering under General Utilities

TL;DR

This paper proves that a broad class of clustering problems, including popular methods like k-means and spectral clustering, are NP-hard, explaining their practical instability and guiding future research towards more stable, interaction-based solutions.

Contribution

It introduces the Universal Clustering Problem (UCP) framework, demonstrating its NP-hardness and unifying diverse clustering paradigms under a common computational complexity barrier.

Findings

UCP is NP-hard via reductions from graph coloring and X3C.

Ten major clustering paradigms are shown to inherit this NP-hardness.

Results explain common failure modes like local optima and greedy traps.

Abstract

Clustering is a central primitive in unsupervised learning, yet practice is dominated by heuristics whose outputs can be unstable and highly sensitive to representations, hyperparameters, and initialisation. Existing theoretical results are largely objective-specific and do not explain these behaviours at a unifying level. We formalise the common optimisation core underlying diverse clustering paradigms by defining the Universal Clustering Problem (UCP): the maximisation of a polynomial-time computable partition utility over a finite metric space. We prove the NP-hardness of UCP via two independent polynomial-time reductions from graph colouring and from exact cover by 3-sets (X3C). By mapping ten major paradigms -- including k-means, GMMs, DBSCAN, spectral clustering, and affinity propagation -- to the UCP framework, we demonstrate that each inherits this fundamental intractability. Our results provide a unified explanation for characteristic failure modes, such as local optima in alternating methods and greedy merge-order traps in hierarchical clustering. Finally, we show that clustering limitations reflect interacting computational and epistemic constraints, motivating a shift toward stability-aware objectives and interaction-driven formulations with explicit guarantees.