In recent years extensions of manifold Ricci curvature to discrete combinatorial objects such as graphs and hypergraphs (popularly called as "network shapes"), have found a plethora of applications in a wide spectrum of research areas ranging over metabolic systems, transcriptional regulatory networks, protein-protein-interaction networks, social networks and brain networks to deep learning models but, in contrast, they have been looked at by relatively fewer researchers in the algorithms and computational complexity community. As an attempt to bring these network Ricci-curvature related problems under the lens of computational complexity and foster further inter-disciplinary interactions, we provide a formal framework for studying algorithmic and computational complexity issues for detecting critical edges in an undirected graph using Ollivier-Ricci curvatures and provide several algorithmic and inapproximability results for problems in this framework. Our results show some interesting connections between our problems, the exact perfect matching and perfect matching blocker problems for bipartite graphs and two well-known combinatorial packing/covering problems.

翻译：近年来，将流形里奇曲率扩展到图与超图等离散组合对象（俗称“网络形状”）的方法，在代谢系统、转录调控网络、蛋白质相互作用网络、社交网络、脑网络乃至深度学习模型等广泛研究领域取得了众多应用。然而，与之相对的是，算法与计算复杂性领域的研究者对此类问题的关注较少。为将这类网络里奇曲率相关问题置于计算复杂性的视角下，并促进进一步的跨学科互动，我们提出一个形式化框架，用于研究利用Ollivier-Ricci曲率检测无向图中关键边的算法与计算复杂性课题，并给出该框架下多个问题的算法结果与不可近似性结论。我们的结果表明，这些问题与二分图上的精确完美匹配及完美匹配阻挡问题，以及两个著名的组合覆盖/打包问题之间存在若干有趣的联系。