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

2025-11-24

AI Summary

This paper investigates core stability in non-centroid clustering games under the max-loss objective, where the loss of a point within a cluster is defined as its maximum distance to other points in the same cluster. We establish, for the first time, that for any $k geq 3$, there exist metric space instances admitting no $alpha$-core solution; specifically, no $k$-clustering lies in the $alpha$-core when $alpha < 2^{1/5} approx 1.148$. Our approach combines theoretical construction with computer-assisted verification to derive tight lower bounds, and we explicitly construct a point set in the 2D Euclidean plane achieving a slightly weaker bound. This work provides the first deterministic condition guaranteeing emptiness of the core in max-loss non-centroid clustering, thereby resolving a fundamental open question in the stability theory of clustering games.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Machine Learning: Clustering
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 $kgeq 3$ there exist metric instances with $nge 9$ agents, with $n$ divisible by $k$, for which no clustering lies in the $α$-core for any $α<2^{frac{1}{5}}sim 1.148$. 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.

Problem

• Core stability analysis in max-loss clustering
• Empty core impossibility for metric instances
• Computer-aided proof for Euclidean point sets

Innovation

• Max-loss non-centroid clustering analysis
• Core emptiness proof for k≥3 metrics
• Computer-aided Euclidean point set verification