Equalizing Closeness Centralities via Edge Additions

TL;DR

This paper explores how to add edges to graphs to make the closeness centralities of two nodes more equal, formalizing two problem variants, analyzing their computational complexity, and providing approximation algorithms where possible.

Contribution

It formalizes two new centrality equalization problems, proves their NP-hardness, and offers approximation algorithms and inapproximability results for these problems.

Findings

Both problems are NP-hard.

A quasilinear-time 6/11-approximation exists for Closeness Ratio Improvement.

Closeness Gap Minimization cannot be approximated multiplicatively unless P=NP.

Abstract

Graph modification problems with the goal of optimizing some measure of a given node's network position have a rich history in the algorithms literature. Less commonly explored are modification problems with the goal of equalizing positions, though this class of problems is well-motivated from the perspective of equalizing social capital, i.e., algorithmic fairness. In this work, we study how to add edges to make the closeness centralities of a given pair of nodes more equal. We formalize two versions of this problem: Closeness Ratio Improvement, which aims to maximize the ratio of closeness centralities between two specified nodes, and Closeness Gap Minimization, which aims to minimize the absolute difference of centralities. We show that both problems are $\textsf{NP}$-hard, and for Closeness Ratio Improvement we present a quasilinear-time $\frac{6}{11}$-approximation, complemented by a bicriteria inapproximability bound. In contrast, we show that Closeness Gap Minimization admits no multiplicative approximation unless $\textsf{P} = \textsf{NP}$. We conclude with a discussion of open directions for this style of problem, including several natural generalizations.