Given a graph $G = (V, E)$ and a model of information flow on that network, a fundamental question is to understand if all the nodes have sufficient access to information generated at other nodes in the graph. If not, we can ask if a small set of edge additions improve information access. Formally, the broadcast value of a network is defined to be the minimum over pairs $u,v \in V$ of the probability that an information cascade starting at $u$ reaches $v$. Recent work in the algorithmic fairness literature has focused on heuristics for adding a few edges to a graph to improve its broadcast. Our goal is to formally study the approximability of the Broadcast Improvement problem: given $G$ and a parameter $k$, find the best set of $k$ edges to add to $G$ in order to maximize the broadcast value of the resulting graph. We develop efficient bicriteria approximation algorithms. If the optimal solution adds $k$ edges and achieves a broadcast of $\beta^*$, we develop algorithms that can (a) add $2k-1$ edges and achieve a broadcast value roughly $(\beta^*)^4$, or (b) add $O(k\log n)$ edges and achieve a broadcast roughly $\beta^*$. We also provide other trade-offs, that can be better depending on $k$ and the parameter associated with propagation in the cascade model. We complement our results by proving that unless P = NP, any algorithm that adds $O(k)$ edges must lose significantly in the approximation of $\beta^*$, resolving an open question. Our techniques are inspired by connections between Broadcast Improvement and problems such as Metric $k$-Center and Diameter Reduction. However, since the objective involves information cascades, we need to develop novel probabilistic tools to reason about the existence of paths in edge-sampled graphs. Finally, we show that our techniques extend to a single-source variant, for which we show both bicriteria algorithms and inapproximability results.

翻译：给定图$G = (V, E)$及其上的信息传播模型，一个核心问题是判断图中所有节点是否对其它节点产生的信息具有充分的访问能力。若否，则可探究是否通过添加少量边能改善信息访问。形式化而言，网络的广播值定义为：对任意节点对$u,v \in V$，从$u$出发的信息级联能到达$v$的最小概率。算法公平性领域的最新研究聚焦于通过启发式方法添加少量边以提升图广播值的实践。本文旨在对广播提升问题的近似性进行形式化研究：给定$G$与参数$k$，寻找最优的$k$条边添加到$G$中，使得新图的广播值最大化。我们提出了高效的双准则近似算法：若最优解添加$k$条边获得广播值$\beta^*$，我们开发的算法可（a）添加$2k-1$条边获得约$(\beta^*)^4$的广播值，或（b）添加$O(k\log n)$条边获得约$\beta^*$的广播值。我们还提供了其他权衡方案，其效果取决于$k$及级联模型中与传播相关的参数。通过证明除非P = NP，否则任何添加$O(k)$条边的算法必然在$\beta^*$的近似度上存在显著损失，我们补充了上述结果，从而解决了一个开放性问题。我们的技术受到广播提升问题与度量$k$-中心及直径缩减等问题之间关联的启发。但由于目标函数涉及信息级联，需要开发新的概率工具来分析边采样图中路径的存在性。最后，我们证明了该技术可扩展至单源变体问题，并为此提出了双准则算法与不可近似性结果。