Given a graph $G = (V, E)$ and a model of information flow on that network, a fundamental question is to understand whether all nodes have sufficient access to information generated at other nodes in the graph. If not, we can ask if a small set of interventions in the form 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 $v$. Having a high broadcast value ensures that every node has sufficient access to information spreading in a network, thus quantifying fairness of access. In this paper, we formally study the Broadcast Improvement problem: given $G$ and a parameter $k$, the goal is to 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 approximation algorithms for this problem. If the optimal solution adds $k$ edges and achieves a broadcast of $\beta^*$, we develop algorithms that can (a) add $k$ edges and achieve a broadcast value roughly $(\beta^*)^4/16^k$, 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 the parameter values. Our algorithms rely on novel probabilistic tools to reason about the existence of paths in edge-sampled graphs, and extend to a single-source variant of the problem, where we obtain analogous algorithmic results. 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 from prior work.

翻译：给定图 $G = (V, E)$ 及其上的信息传播模型，一个基本问题是理解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，我们可以探讨是否通过少量以边添加形式进行的干预能够改善信息访问。形式上，网络的广播值定义为从节点 $u$ 出发的信息级联到达节点 $v$ 的概率在所有节点对 $u,v \in V$ 上的最小值。较高的广播值能确保每个节点对网络中传播的信息具有充分访问能力，从而量化了访问的公平性。本文对广播改善问题展开形式化研究：给定 $G$ 与参数 $k$，目标是找到最优的 $k$ 条边添加到 $G$ 中，以使所得图的广播值最大化。我们针对该问题开发了高效的近似算法。若最优解添加 $k$ 条边并达到广播值 $\beta^*$，我们开发的算法能够：(a) 添加 $k$ 条边并实现约 $(\beta^*)^4/16^k$ 的广播值，或 (b) 添加 $O(k\log n)$ 条边并实现约 $\beta^*$ 的广播值。我们还提供了其他权衡方案，这些方案根据参数取值可能具有更优性能。我们的算法依赖于新颖的概率工具来推理边采样图中路径的存在性，并可推广至该问题的单源变体，其中我们获得了类似的算法结果。我们通过证明除非 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，从而补充了我们的结果，这解决了先前工作中悬而未决的问题。