Correlation Clustering Generalized

TL;DR

This paper advances correlation clustering by analyzing LP relaxations, improving approximation algorithms for LambdaCC and MotifCC, and extending results to hypergraph settings with better approximation factors.

Contribution

It provides new approximation algorithms and bounds for LambdaCC and MotifCC, including hypergraph generalizations and parameter-independent results.

Findings

LP relaxation of LambdaCC has a (\,log n) integrality gap.

Improved approximation factor from 9 to 8 for hyperedges of degree 3 in MotifCC.

General approximation of 4(k-1) for hyperedges with maximum degree k.

Abstract

We present new results for LambdaCC and MotifCC, two recently introduced variants of the well-studied correlation clustering problem. Both variants are motivated by applications to network analysis and community detection, and have non-trivial approximation algorithms. We first show that the standard linear programming relaxation of LambdaCC has a $\Theta(\log n)$ integrality gap for a certain choice of the parameter $\lambda$. This sheds light on previous challenges encountered in obtaining parameter-independent approximation results for LambdaCC. We generalize a previous constant-factor algorithm to provide the best results, from the LP-rounding approach, for an extended range of $\lambda$. MotifCC generalizes correlation clustering to the hypergraph setting. In the case of hyperedges of degree $3$ with weights satisfying probability constraints, we improve the best approximation factor from $9$ to $8$. We show that in general our algorithm gives a $4(k-1)$ approximation when hyperedges have maximum degree $k$ and probability weights. We additionally present approximation results for LambdaCC and MotifCC where we restrict to forming only two clusters.