The Core in Max-Loss Non-Centroid Clustering Can Be Empty

TL;DR

This paper demonstrates that in max-loss non-centroid clustering, the core can be empty, providing tight bounds and computer-aided proofs, marking a novel impossibility result in clustering stability.

Contribution

It proves the existence of metric instances where the core is empty for all clusterings under the max-loss objective, a first in this research area.

Findings

Core can be empty in non-centroid clustering under max-loss.

Established tight bounds for core stability.

Used computer-aided proofs to identify specific point sets.

Abstract

We study core stability in non-centroid clustering under the max-loss objective, where each agent's loss is the maximum distance to other members of their cluster. We prove that for all $k\geq 3$ there exist metric instances with $n\ge 9$ agents, with $n$ divisible by $k$, for which no clustering lies in the $\alpha$-core for any $\alpha<2^{\frac{1}{5}}\sim 1.148$. The bound is tight for our construction. Using a computer-aided proof, we also identify a two-dimensional Euclidean point set whose associated lower bound is slightly smaller than that of our general construction. This is, to our knowledge, the first impossibility result showing that the core can be empty in non-centroid clustering under the max-loss objective.