Correlation Clustering and Biclustering with Locally Bounded Errors

TL;DR

This paper introduces a flexible framework for correlation clustering that minimizes locally bounded errors, providing approximation algorithms that work for various error-based objectives.

Contribution

It generalizes correlation clustering by allowing customizable error functions and offers a rounding algorithm with constant-factor approximation guarantees.

Findings

Provides a new framework for locally bounded error objectives.

Develops a rounding algorithm with constant-factor approximation.

Applies to a wide range of error-based clustering objectives.

Abstract

We consider a generalized version of the correlation clustering problem, defined as follows. Given a complete graph $G$ whose edges are labeled with $+$ or $-$, we wish to partition the graph into clusters while trying to avoid errors: $+$ edges between clusters or $-$ edges within clusters. Classically, one seeks to minimize the total number of such errors. We introduce a new framework that allows the objective to be a more general function of the number of errors at each vertex (for example, we may wish to minimize the number of errors at the worst vertex) and provide a rounding algorithm which converts "fractional clusterings" into discrete clusterings while causing only a constant-factor blowup in the number of errors at each vertex. This rounding algorithm yields constant-factor approximation algorithms for the discrete problem under a wide variety of objective functions.