Expander Pruning with Polylogarithmic Worst-Case Recourse and Update Time

Expander graphs are known to be robust to edge deletions in the following sense: for any online sequence of edge deletions $e_1, e_2, \ldots, e_k$ to an $m$-edge graph $G$ that is initially a $\phi$-expander, the algorithm can grow a set $P \subseteq V$ such that at any time $t$, $G[V \setminus P]$ is an expander of the same quality as the initial graph $G$ up to a constant factor and the set $P$ has volume at most $O(t/\phi)$. However, currently, there is no algorithm to grow $P$ with low worst-case recourse that achieves any non-trivial guarantee. In this work, we present an algorithm that achieves near-optimal guarantees: we give an algorithm that grows $P$ only by $\tilde{O}(1/\phi^2)$ vertices per time step and ensures that $G[V \setminus P]$ remains $\tilde{\Omega}(\phi)$-expander at any time. Even more excitingly, our algorithm is extremely efficient: it can process each update in near-optimal worst-case update time $\tilde{O}(1/\phi^2)$. This affirmatively answers the main open question posed in [SW19] whether such an algorithm exists. By combining our results with recent techniques in [BvdBPG+22], we obtain the first adaptive algorithms to maintain spanners, cut and spectral sparsifiers with $\tilde{O}(n)$ edges and polylogarithmic approximation guarantees, worst-case update time and recourse. More generally, we believe that worst-case pruning is an essential tool for obtaining worst-case guarantees in dynamic graph algorithms and online algorithms.