Maintaining Expander Decompositions via Sparse Cuts

In this article, we show that the algorithm of maintaining expander decompositions in graphs undergoing edge deletions directly by removing sparse cuts repeatedly can be made efficient. Formally, for an $m$-edge undirected graph $G$, we say a cut $(S, \overline{S})$ is $\phi$-sparse if $|E_G(S, \overline{S})| < \phi \cdot \min\{vol_G(S), vol_G(\overline{S})\}$. A $\phi$-expander decomposition of $G$ is a partition of $V$ into sets $X_1, X_2, \ldots, X_k$ such that each cluster $G[X_i]$ contains no $\phi$-sparse cut (meaning it is a $\phi$-expander) with $\tilde{O}(\phi m)$ edges crossing between clusters. A natural way to compute a $\phi$-expander decomposition is to decompose clusters by $\phi$-sparse cuts until no such cut is contained in any cluster. We show that even in graphs undergoing edge deletions, a slight relaxation of this meta-algorithm can be implemented efficiently with amortized update time $m^{o(1)}/\phi^2$. Our approach naturally extends to maintaining directed $\phi$-expander decompositions and $\phi$-expander hierarchies and thus gives a unifying framework while having simpler proofs than previous state-of-the-art work. In all settings, our algorithm matches the run-times of previous algorithms up to subpolynomial factors. Moreover, our algorithm provides stronger guarantees for $\phi$-expander decompositions. For example, for graphs undergoing edge deletions, our approach is the first to maintain a dynamic expander decomposition where each updated decomposition is a refinement of the previous decomposition, and our approach is the first to guarantee a sublinear $\phi m^{1+o(1)}$ bound on the total number of edges that cross between clusters across the entire sequence of dynamic updates.