Learning Augmented Graph $k$-Clustering

TL;DR

This paper extends learning-augmented $k$-clustering to general metric spaces, relaxes cluster size constraints, and establishes query complexity lower bounds, enhancing theoretical understanding and practical flexibility.

Contribution

It generalizes learning-augmented $k$-clustering to arbitrary metrics, relaxes cluster size restrictions, and proves query complexity bounds under ETH.

Findings

Applicable to graph-structured and non-Euclidean data

Relaxed cluster size constraints for imbalanced datasets

Established query complexity lower bounds under ETH

Abstract

Clustering is a fundamental task in unsupervised learning. Previous research has focused on learning-augmented $k$-means in Euclidean metrics, limiting its applicability to complex data representations. In this paper, we generalize learning-augmented $k$-clustering to operate on general metrics, enabling its application to graph-structured and non-Euclidean domains. Our framework also relaxes restrictive cluster size constraints, providing greater flexibility for datasets with imbalanced or unknown cluster distributions. Furthermore, we extend the hardness of query complexity to general metrics: under the Exponential Time Hypothesis (ETH), we show that any polynomial-time algorithm must perform approximately $\Omega(k / \alpha)$ queries to achieve a $(1 + \alpha)$-approximation. These contributions strengthen both the theoretical foundations and practical applicability of learning-augmented clustering, bridging gaps between traditional methods and real-world challenges.