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 and quantum computing 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 the exact perfect matching and perfect matching blocker problems for bipartite graphs and our problems.

翻译：近年来，流形里奇曲率向离散组合对象（如图与超图，常被称为“网络形状”）的扩展，已在代谢系统、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与量子计算等广泛研究领域中得到大量应用。然而相比之下，算法与计算复杂性领域的研究者对此关注相对较少。为将此类网络里奇曲率相关问题纳入计算复杂性研究范畴并促进跨学科交流，本文提出一个形式化框架，用于研究基于Ollivier-Ricci曲率检测无向图中关键边的算法与计算复杂性问题，并在此框架下给出若干算法设计与不可近似性结果。我们的研究揭示了二分图的精确完美匹配问题、完美匹配阻塞问题与本问题之间若干有趣的理论联系。