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

Robert Bredereck, Eva Deltl, Leon Kellerhals, Jannik Peters

TL;DR

This paper proves that the core can be empty in non-centroid clustering under the max-loss objective by constructing a metric-space gadget for any $k \ge 3$ and $n \ge 9$ with $k \mid n$ that admits no $k$-clustering in the $\alpha$-core for any $\alpha < 2^{\frac{1}{5}}$. It further enhances the result with a computer-aided Euclidean counterexample yielding a slightly smaller bound on $\alpha$, suggesting the bound is tight for their construction. The result provides the first impossibility demonstration of an empty core in this setting, with extensions to arbitrary $k$ via dummy clusters and a concrete path toward understanding core stability in non-centroid, max-loss clustering. The work motivates questions about potential improvements of the bound and about Euclidean versus general metric-space behavior in core guarantees.

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 $α$-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.

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

TL;DR

This paper proves that the core can be empty in non-centroid clustering under the max-loss objective by constructing a metric-space gadget for any and with that admits no -clustering in the -core for any . It further enhances the result with a computer-aided Euclidean counterexample yielding a slightly smaller bound on , suggesting the bound is tight for their construction. The result provides the first impossibility demonstration of an empty core in this setting, with extensions to arbitrary via dummy clusters and a concrete path toward understanding core stability in non-centroid, max-loss clustering. The work motivates questions about potential improvements of the bound and about Euclidean versus general metric-space behavior in core guarantees.

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 there exist metric instances with agents, with divisible by , for which no clustering lies in the -core for any . 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.

Paper Structure

This paper contains 7 sections, 2 theorems, 6 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation.
  3. Proof of \ref{['thm:nope']}
  4. Proof of \ref{['thm:nope']}
  5. Case 2: There is a mixed cluster.
  6. Outlook
  7. Points for \ref{['thm:euclnope']}

Key Result

Theorem 1

For every $k \ge 3$ there exists a metric space with $n \ge 9$ agents, where $n$ is divisible by $k$, for which no $k$-clustering is in the $\alpha$-core under the max-loss objective, for any $\alpha < 2^{\frac{1}{5}} < 1.149$.

Figures (2)

  • Figure 1: Counterexample for non-centroid max-loss: edge labels specify their length; the distance between any two points is the length of the shortest path.
  • Figure 2: The points for the counterexample in two-dimensional Euclidean space (rotated by 90 degrees). $w$ was placed closer to the remaining points for an easier illustration. The black ellipses depict the clustering that is in the $1.139$-core and the blue (dashed) ellipse is the $1.139$-blocking coalition.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2