Fully Dynamic MIS in Uniformly Sparse Graphs

TL;DR

This paper presents an improved algorithm for maintaining a maximal independent set in dynamically changing sparse graphs, achieving faster update times especially for graphs with low arboricity, including many real-world graph classes.

Contribution

The paper introduces a significantly faster dynamic MIS algorithm for uniformly sparse graphs with low arboricity, improving upon previous sublinear update time results.

Findings

Amortized update time is $O(lpha^2  log^2 n)$ for graphs with arboricity lpha.

Update time is polylogarithmic for low arboricity graphs, including minor-free and real-world graphs.

The algorithm outperforms previous methods for all graphs with arboricity up to $m^{3/8 - psilon}$.

Abstract

We consider the problem of maintaining a maximal independent set (MIS) in a dynamic graph subject to edge insertions and deletions. Recently, Assadi, Onak, Schieber and Solomon (STOC 2018) showed that an MIS can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this paper we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity $\alpha$, the amortized update time of our algorithm is $O(\alpha^2 \cdot \log^2 n)$, where $n$ is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs as well as some classes of `real world' graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by $m^{3/8 - \epsilon}$, for any constant $\epsilon > 0$. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed $m^{1/2}$.