Min-Max Correlation Clustering via Neighborhood Similarity

TL;DR

This paper introduces a nearly linear time, $(3 + \\epsilon)$-approximation algorithm for min-max correlation clustering on complete graphs, improving previous guarantees and extending to parallel and streaming models.

Contribution

The paper presents a novel combinatorial algorithm with improved approximation ratio and efficiency for min-max correlation clustering, applicable in parallel and streaming settings.

Findings

Achieves a $(3 + \\epsilon)$-approximation in nearly linear time.

Extends the algorithm to MPC and semi-streaming models.

Uses neighborhood similarity queries and random projection for efficiency.

Abstract

We present an efficient algorithm for the min-max correlation clustering problem. The input is a complete graph where edges are labeled as either positive $(+)$ or negative $(-)$, and the objective is to find a clustering that minimizes the $\ell_{\infty}$-norm of the disagreement vector over all vertices. We resolve this problem with an efficient $(3 + \epsilon)$-approximation algorithm that runs in nearly linear time, $\tilde{O}(|E^+|)$, where $|E^+|$ denotes the number of positive edges. This improves upon the previous best-known approximation guarantee of 4 by Heidrich, Irmai, and Andres, whose algorithm runs in $O(|V|^2 + |V| D^2)$ time, where $|V|$ is the number of nodes and $D$ is the maximum degree in the graph. Furthermore, we extend our algorithm to the massively parallel computation (MPC) model and the semi-streaming model. In the MPC model, our algorithm runs on machines with memory sublinear in the number of nodes and takes $O(1)$ rounds. In the streaming model, our algorithm requires only $\tilde{O}(|V|)$ space, where $|V|$ is the number of vertices in the graph. Our algorithms are purely combinatorial. They are based on a novel structural observation about the optimal min-max instance, which enables the construction of a $(3 + \epsilon)$-approximation algorithm using $O(|E^+|)$ neighborhood similarity queries. By leveraging random projection, we further show these queries can be computed in nearly linear time.