Towards a New Theory of Optimal Dynamic Graph Algorithms

Introduction

Dynamic graph algorithms are of increasing critical importance. They are crucial for coping with dynamic networks, which model the ever-changing physical world, and have been instrumental in achieving numerous major breakthroughs in static graph algorithms.

Research Goals

The holy grail in the field of dynamic graph algorithms has been to design algorithms with poly-logarithmic (in the input size) update time. However, recent exciting developments, in which the PI has played a central role, aim to push the update time toward an absolute constant independent of the input size – which is qualitatively very different than a poly-log bound.

This goal is of fundamental importance not just from a theoretical perspective, but also from a practical viewpoint, due to the rapidly growing size of modern networks.

Intrinsically Optimal Algorithms

An algorithm is intrinsically optimal if its update time matches the ratio of the problem’s static time complexity to the input size. The main question underlying this research is:

• Which graph problems admit intrinsically optimal update time?

Only a few intrinsically optimal graph algorithms are known. The unique goal of this project is to establish a systematic study of intrinsically optimal algorithms.

Provably Optimal Algorithms

We will also study provably optimal algorithms, aiming to advance our understanding of the thin line that separates these two distinct optimality notions. To achieve this goal, we must go far beyond the current state-of-the-art, and in particular, confront some of the most central problems in the field.

Impact and Significance

Meeting the project’s main goal, even partially, will be groundbreaking. Results of this project will facilitate the use of dynamic algorithms in real-world application domains and will also be illuminating to other fields, such as distributed computing and fine-grained complexity.

Consequently, we believe this research has the potential of revolutionizing the field of dynamic graph algorithms and impacting related fields, thus enriching the general landscape of computer science.