The Power of Vertex Sparsifiers in Dynamic Graph Algorithms

TL;DR

This paper introduces a novel framework leveraging vertex sparsification to develop efficient dynamic graph algorithms for minor-free graphs, achieving improved update and query times for problems like electrical flow energy, max flow, and shortest paths.

Contribution

It establishes a new connection between dynamic graph algorithms and vertex sparsification, providing the first systematic approach with concrete algorithms and bounds for minor-free graphs.

Findings

Developed a Monte Carlo randomized algorithm for electrical flow approximation with sublinear update time.

Extended the framework to handle minor-free graphs with similar efficiency guarantees.

Provided bounds for maintaining electrical flow energy in incremental subgraph models.

Abstract

We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms. Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for $(1+\varepsilon)$-approximating the energy of electrical flows in $n$-vertex planar graphs with $\tilde{O}(r\varepsilon^{-2})$ worst-case update time and $\tilde{O}((r+\frac{n}{\sqrt{r}})\varepsilon^{-2})$ worst-case query time, for any $r$ larger than some constant. For $r=n^{2/3}$, this gives $\tilde{O}(n^{2/3}\varepsilon^{-2})$ update time and $\tilde{O}(n^{2/3}\varepsilon^{-2})$ query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification. We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest.