Expander Pruning with Polylogarithmic Worst-Case Recourse and Update Time

2025-04-01

AI Summary

This paper studies the online graph pruning problem under dynamic edge deletions while preserving expansion: given a φ-expander graph, maintain a pruned vertex set (P subseteq V) such that the residual subgraph (G[V setminus P]) remains an (Omega(phi))-expander for any adversarial sequence of edge deletions, with controlled growth of (|P|). We present the first worst-case nearly optimal algorithm, achieving ( ilde{O}(1/phi^2)) update time and vertex adjustment per step—resolving the open problem posed by Sawlani & Wang (SW19). Technically, our approach integrates spectral analysis of expanders, local repair mechanisms, hierarchical sampling, and a dynamic sparsification framework. As a consequence, we obtain the first worst-case efficient algorithms for maintaining adaptive spanners and cut/spectral sparsifiers: size ( ilde{O}(n)), polylogarithmic stretch/spectral approximation, and ( ilde{O}(1/phi^2)) amortized update time.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $phi$-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 $ ilde{O}(1/phi^2)$ vertices per time step and ensures that $G[V setminus P]$ remains $ ilde{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 $ ilde{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 $ ilde{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.

Problem

• Maintain expander properties after edge deletions efficiently
• Achieve low worst-case recourse and update time
• Develop adaptive algorithms for dynamic graph structures

Innovation

• Polylogarithmic worst-case recourse and update time
• Maintains expander properties with minimal vertex growth
• Efficient dynamic graph algorithms with near-optimal guarantees