Characterizing shapes of high-dimensional objects via Ricci curvatures plays a critical role in many research areas in mathematics and physics. However, even though several discretizations of Ricci curvatures for discrete combinatorial objects such as networks have been proposed and studied by mathematicians, the computational complexity aspects of these discretizations have escaped the attention of theoretical computer scientists to a large extent. In this paper, we study one such discretization, namely the Ollivier-Ricci curvature, from the perspective of efficient computation by fine-grained reductions and local query-based algorithms. Our main contributions are the following. (a) We relate our curvature computation problem to minimum weight perfect matching problem on complete bipartite graphs via fine-grained reduction. (b) We formalize the computational aspects of the curvature computation problems in suitable frameworks so that they can be studied by researchers in local algorithms. (c) We provide the first known lower and upper bounds on queries for query-based algorithms for the curvature computation problems in our local algorithms framework. En route, we also illustrate a localized version of our fine-grained reduction. We believe that our results bring forth an intriguing set of research questions, motivated both in theory and practice, regarding designing efficient algorithms for curvatures of objects.

翻译：通过Ricci曲率刻画高维物体的形状在数学和物理学的许多研究领域中起着关键作用。然而，尽管数学家们已经提出并研究了多种针对网络等离散组合对象的Ricci曲率离散化方案，但这些离散化方法的计算复杂性在很大程度上未引起理论计算机科学家的关注。本文从细粒度归约和基于局部查询的算法视角，研究其中一种离散化方案——Ollivier-Ricci曲率的高效计算问题。我们的主要贡献如下：(a) 通过细粒度归约，将曲率计算问题与完全二分图上的最小权完美匹配问题建立关联；(b) 在合适的框架内形式化曲率计算问题的计算层面，使其可供局部算法领域的研究者研究；(c) 在局部算法框架下，首次给出曲率计算问题基于查询算法所需查询次数的已知下界与上界。在此过程中，我们还阐明了细粒度归约的局部化版本。我们相信，这些成果为设计物体曲率的高效算法——这一兼具理论与实践动机的研究课题——提出了一系列引人入胜的研究问题。