Equalizing Closeness Centralities via Edge Additions

TL;DR

The paper studies equalizing two specified nodes' closeness centralities via edge additions. It formalizes two problems, Closeness Ratio Improvement and Closeness Gap Minimization, and shows both are NP-hard, with the ratio variant admitting a quasilinear-time $\frac{6}{11}$-approximation and bicriteria inapproximability, while the gap variant has no multiplicative approximation unless $P=NP$. The authors develop structural lemmas and an efficient algorithm that uses edges from $a$ to the neighborhood of $b$ to achieve the $\frac{6}{11}$-approximation, with runtime $O((n+m)\log k)$. They also establish strong hardness results, including a Set Cover-based reduction, and discuss natural generalizations to group fairness and all-pairs balancing, outlining directions for future work in algorithmic fairness for graph modification.

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.

Figures (4)

• Figure 1: The constructions used for counterexamples in Section \ref{['sec:strategies']}. Dashed edges are potential additions, squiggly lines are paths of length $d$, and $IS_X$ denotes an independent set of size $X$.
• Figure 2: The worst possible closeness ratio when $a$ and $b$ are adjacent.
• Figure 3: The construction given by \ref{['t1-hard']}. We denote set vertices by $s_j$ and element vertices by $v_i$. Here, $IS_X$ is an independent set of $X$ vertices, with each vertex adjacent to $a$. We refer to the proof of \ref{['t1-hard']} for a formal description of the construction and the accompanying analysis.
• Figure 4: A depiction of the construction given by \ref{['thm:inapprox']}. Given an instance of Set Cover, set vertices $s_j$ correspond to sets. Each element vertex $v_i$ is duplicated into $m$ twins. Here, $IS_X$ denotes an independent set of size $mn+ck$, with each vertex adjacent to $a$. We refer to the proof of \ref{['thm:inapprox']} for a formal description of the construction and the accompanying analysis.

Theorems & Definitions (33)

• Theorem 1
• proof
• Claim 1
• proof
• Corollary 1
• proof
• Theorem 2
• proof
• Claim 2
• proof