Inapproximability for Local Correlation Clustering and Dissimilarity Hierarchical Clustering

TL;DR

This paper establishes hardness of approximation results for local correlation clustering and dissimilarity-based hierarchical clustering, showing they are difficult to approximate within specific constant factors under common complexity assumptions.

Contribution

It provides the first APX-hardness bounds for these clustering problems, advancing understanding of their computational complexity.

Findings

Correlation Clustering with local objectives is hard to approximate within 4/3.

Hierarchical Clustering with dissimilarity maximization is hard to approximate within ~0.9967.

Hardness results are based on P vs NP and the Unique Games Conjecture.

Abstract

We present hardness of approximation results for Correlation Clustering with local objectives and for Hierarchical Clustering with dissimilarity information. For the former, we study the local objective of Puleo and Milenkovic (ICML '16) that prioritizes reducing the disagreements at data points that are worst off and for the latter we study the maximization version of Dasgupta's cost function (STOC '16). Our APX hardness results imply that the two problems are hard to approximate within a constant of 4/3 ~ 1.33 (assuming P vs NP) and 9159/9189 ~ 0.9967 (assuming the Unique Games Conjecture) respectively.