DeepAI AI Chat
Log In Sign Up

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

by   Bhaskar DasGupta, et al.

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.


page 1

page 2

page 3

page 4


Parameterized Fine-Grained Reductions

During recent years the field of fine-grained complexity has bloomed to ...

New Techniques for Proving Fine-Grained Average-Case Hardness

The recent emergence of fine-grained cryptography strongly motivates dev...

Fine-Grained I/O Complexity via Reductions: New lower bounds, faster algorithms, and a time hierarchy

This paper initiates the study of I/O algorithms (minimizing cache misse...

Worst-Case to Expander-Case Reductions

In recent years, the expander decomposition method was used to develop m...

Fine-Grained Complexity of Regular Path Queries

A regular path query (RPQ) is a regular expression q that returns all no...