The price of incrementality in k-center clustering

2026-06-07

AI Summary

This study addresses the incremental k-center clustering problem, where centers must be output one by one such that every prefix of the solution maintains a provable approximation guarantee. By constructing a lower bound in metric spaces, the work establishes—for the first time—that the incremental requirement itself inherently limits the achievable approximation ratio to no better than 2, even without computational constraints. This result reveals the fundamental performance cost imposed by incrementality and delineates the theoretical limits of the problem. The analysis combines geometric constructions with lower-bound proof techniques, introducing an innovative method for designing adversarial instances that simultaneously constrain all levels of the clustering hierarchy.

Abstract

The $k$-center problem is one of the best-studied and most intuitive clustering formulations. It asks, given a set of $n$ points in a metric space, for $k$ of the points to be designated as cluster centers, so that the maximum distance of an input point to its nearest center is minimized. Gonzalez's greedy algorithm from 1985 is a simple and efficient way to find a $2$-approximate solution. The algorithm has the attractive feature of \emph{incrementality}: it outputs the centers one by one, with a guaranteed $2$-approximation for every prefix of the obtained sequence of centers. Incrementality imposes a geometric constraint on how solutions can be built, and it is natural to ask whether this comes at a price in the quality of the solution. It is known that in polynomial time, the approximation ratio of $2$ is best possible, assuming $P \neq NP$. In this paper we show that even with \emph{unlimited} computational power, the factor $2$ cannot be improved, if the solution is required to be built incrementally. The lower bound construction imposes a tradeoff between all $n$ levels of the clustering simultaneously; it was obtained with the help of ChatGPT, an aspect we discuss in Section 3 of the paper.

Problem

• k-center clustering
• incrementality
• approximation ratio
• lower bound
• computational hardness

Innovation

• k-center clustering
• incrementality
• approximation lower bound
• greedy algorithm
• computational hardness