Universal NP-Hardness of Clustering under General Utilities

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.

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