Correlation Clustering with Global Weight Bounds

Abstract

Given a set of objects and nonnegative real weights expressing “positive” and “negative” feeling of clustering any two objects together, min-disagreement correlation clustering partitions the input object set so as to minimize the sum of the intra-cluster negative-type weights plus the sum of the inter-cluster positive-type weights. Min-disagreement correlation clustering is \(\mathbf {APX}\)-hard, but efficient constant-factor approximation algorithms exist if the weights are bounded in some way. The weight bounds so far studied in the related literature are mostly local, as they are required to hold for every object-pair. In this paper, we introduce the problem of min-disagreement correlation clustering with global weight bounds, i.e., constraints to be satisfied by the input weights altogether. Our main result is a sufficient condition that establishes when any algorithm achieving a certain approximation under the probability constraint keeps the same guarantee on an input that violates the constraint. This extends the range of applicability of the most prominent existing correlation-clustering algorithms, including the popular Pivot, thus providing benefits, both theoretical and practical. Experiments demonstrate the usefulness of our approach, in terms of both worthiness of employing existing efficient algorithms, and guidance on the definition of weights from feature vectors in a task of fair clustering.