Modification-Fair Cluster Editing

TL;DR

This paper introduces a modification-fairness constraint to the Cluster Editing problem, ensuring subgroup edits are proportional to their size, and demonstrates that the problem remains computationally manageable with minimal impact on solution cost.

Contribution

It formalizes a new fairness constraint for Cluster Editing, analyzes its computational complexity, and empirically shows low cost for fair solutions on social network data.

Findings

Modification-fair Cluster Editing is NP-hard even with limited edge insertions.

The problem remains fixed-parameter tractable with respect to the number of edits.

Fair solutions incur only a small additional cost compared to non-fair solutions.

Abstract

The classic Cluster Editing problem (also known as Correlation Clustering) asks to transform a given graph into a disjoint union of cliques (clusters) by a small number of edge modifications. When applied to vertex-colored graphs (the colors representing subgroups), standard algorithms for the NP-hard Cluster Editing problem may yield solutions that are biased towards subgroups of data (e.g., demographic groups), measured in the number of modifications incident to the members of the subgroups. We propose a modification fairness constraint which ensures that the number of edits incident to each subgroup is proportional to its size. To start with, we study Modification-Fair Cluster Editing for graphs with two vertex colors. We show that the problem is NP-hard even if one may only insert edges within a subgroup; note that in the classic "non-fair" setting, this case is trivially polynomial-time solvable. However, in the more general editing form, the modification-fair variant remains fixed-parameter tractable with respect to the number of edge edits. We complement these and further theoretical results with an empirical analysis of our model on real-world social networks where we find that the price of modification-fairness is surprisingly low, that is, the cost of optimal modification-fair solutions differs from the cost of optimal "non-fair" solutions only by a small percentage.