On computing Discretized Ricci curvatures of graphs: local algorithms and (localized) fine-grained reductions

TL;DR

This paper investigates the computational complexity of discretized Ricci curvature, specifically Ollivier-Ricci curvature, on graphs, introducing local algorithms and reductions to improve understanding of efficient computation methods.

Contribution

It relates curvature computation to minimum weight perfect matching, formalizes the problem for local algorithms, and establishes bounds on query complexity for these algorithms.

Findings

Connected curvature computation to perfect matching problem.

Formalized curvature computation in local algorithms framework.

Established bounds on query complexity for curvature algorithms.

Abstract

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.