Fully Dynamic Spectral and Cut Sparsifiers for Directed Graphs

TL;DR

This paper introduces new algorithms for fully dynamic spectral and cut sparsification of directed graphs, achieving efficient update times and maintaining sparsifiers with provable guarantees, advancing the understanding of graph sparsification in dynamic settings.

Contribution

It proposes the first fully dynamic algorithms for spectral and cut sparsifiers in directed graphs with polylogarithmic update times and introduces a new degree-balance preserving spectral approximation notion.

Findings

Achieves amortized update time of O(ε^{-2} polylog(n)) for spectral sparsifiers.

Maintains constant-factor approximation sparsifiers for partially symmetrized graphs.

Develops a fully dynamic cut sparsifier for β-balanced directed graphs with polylogarithmic update time.

Abstract

Recent years have seen extensive research on directed graph sparsification. In this work, we initiate the study of fast fully dynamic spectral and cut sparsification algorithms for directed graphs. We introduce a new notion of spectral sparsification called degree-balance preserving spectral approximation, which maintains the difference between the in-degree and out-degree of each vertex. The approximation error is measured with respect to the corresponding undirected Laplacian. This notion is equivalent to direct Eulerian spectral approximation when the input graph is Eulerian. Our algorithm achieves an amortized update time of $O(\varepsilon^{-2} \cdot \text{polylog}(n))$ and produces a sparsifier of size $O(\varepsilon^{-2} n \cdot \text{polylog}(n))$. Additionally, we present an algorithm that maintains a constant-factor approximation sparsifier of size $O(n \cdot \text{polylog}(n))$ against an adaptive adversary for $O(\text{polylog}(n))$-partially symmetrized graphs, a notion introduced in [Kyng-Meierhans-Probst Gutenberg '22]. A $\beta$-partial symmetrization of a directed graph $\vec{G}$ is the union of $\vec{G}$ and $\beta \cdot G$, where $G$ is the corresponding undirected graph of $\vec{G}$. This algorithm also achieves a polylogarithmic amortized update time. Moreover, we develop a fully dynamic algorithm for maintaining a cut sparsifier for $\beta$-balanced directed graphs, where the ratio between weighted incoming and outgoing edges of any cut is at most $\beta$. This algorithm explicitly maintains a cut sparsifier of size $O(\varepsilon^{-2}\beta n \cdot \text{polylog}(n))$ in worst-case update time $O(\varepsilon^{-2}\beta \cdot \text{polylog}(n))$.