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arXiv · 2504.00544

Expander Pruning with Polylogarithmic Worst-Case Recourse and Update Time

Simon Meierhans·Maximilian Probst Gutenberg·Thatchaphol Saranurak
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Abstract

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 $ϕ$-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 $O(t/ϕ)$. 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 $\tilde{O}(1/ϕ^2)$ vertices per time step and ensures that $G[V \setminus P]$ remains $\tildeΩ(ϕ)$-expander at any time. Even more excitingly, our algorithm is extremely efficient: it can process each update in near-optimal worst-case update time $\tilde{O}(1/ϕ^2)$. 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.

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Simon Meierhans, Maximilian Probst Gutenberg, Thatchaphol Saranurak. 2025-04-01. Expander Pruning with Polylogarithmic Worst-Case Recourse and Update Time. https://arxiv.org/abs/2504.00544

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