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Expander Pruning with Polylogarithmic Worst-Case Recourse and Update Time

Simon Meierhans, Maximilian Probst Gutenberg, Thatchaphol Saranurak

TL;DR

This work addresses the problem of maintaining expansion in a graph undergoing online edge deletions by pruning a small, monotonically growing vertex set. The authors develop a deterministic framework combining batch pruning, flow certificates, and fast flow/backtracking techniques to achieve near-optimal worst-case recourse and update time, specifically $ ilde{O}(1/ ilde{φ}^2)$ per update and $ ilde{O}(1/ ilde{φ}^2)$ growth in the pruned set per step, while preserving a $ ilde{Ω}( ilde{φ})$-expander in the remaining graph. They further extend these ideas to yield first adaptive algorithms for spanners, cut and spectral sparsifiers with $ ilde{O}(n)$ edges and polylogarithmic guarantees under worst-case updates, and they connect these pruning techniques to fully-dynamic connectivity and online algorithm design. The introduced flow certificates, batching schemes, and backtracking data structures form a versatile toolkit for obtaining worst-case guarantees in dynamic graph problems, with potential broad impact on online computation and network design.

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.

Expander Pruning with Polylogarithmic Worst-Case Recourse and Update Time

TL;DR

This work addresses the problem of maintaining expansion in a graph undergoing online edge deletions by pruning a small, monotonically growing vertex set. The authors develop a deterministic framework combining batch pruning, flow certificates, and fast flow/backtracking techniques to achieve near-optimal worst-case recourse and update time, specifically per update and growth in the pruned set per step, while preserving a -expander in the remaining graph. They further extend these ideas to yield first adaptive algorithms for spanners, cut and spectral sparsifiers with edges and polylogarithmic guarantees under worst-case updates, and they connect these pruning techniques to fully-dynamic connectivity and online algorithm design. The introduced flow certificates, batching schemes, and backtracking data structures form a versatile toolkit for obtaining worst-case guarantees in dynamic graph problems, with potential broad impact on online computation and network design.

Abstract

Expander graphs are known to be robust to edge deletions in the following sense: for any online sequence of edge deletions to an -edge graph that is initially a -expander, the algorithm can grow a set such that at any time , is an expander of the same quality as the initial graph up to a constant factor and the set has volume at most . However, currently, there is no algorithm to grow 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 only by vertices per time step and ensures that remains -expander at any time. Even more excitingly, our algorithm is extremely efficient: it can process each update in near-optimal worst-case update time . 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 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.

Paper Structure

This paper contains 48 sections, 14 theorems, 12 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Applications in Dynamic Graph Algorithms.
  4. Fully-Dynamic Connectivity with Worst-Case Time via Expander Pruning.
  5. Applications in Online Algorithms.
  6. Overview
  7. Review: Expander Pruning with Low Amortized Recourse.
  8. Flow Certificates for Refined Expander Pruning.
  9. Expanders are Robust, Even if the Amortized Pruning Suggests Otherwise.
  10. Relaxing the Resting Property via Batching and Flow Certificate Composition.
  11. An Amortized Online Algorithm with Resting Periods via Increasing Sink Capacities.
  12. Fast Flow Backtracking.
  13. Fast Flow Computations.
  14. Expander Pruning with Worst-Case Update Time and Recourse.
  15. Preliminaries
  16. ...and 33 more sections

Key Result

Theorem 1.3

Given an $m$-edge $\phi$-expander graph $G = (V, E)$ and a sequence of up to $\tilde{\Omega}(\phi \cdot m)$ edge deletions to $G$. There is a deterministic algorithm that processes each edge deletion in time $\tilde{O}(1/\phi^2)$ and adds at most $\tilde{O}(1/\phi^2)$ vertices to the initially empt

Figures (2)

  • Figure 1: On the left: After deleting the red edges and adding $8/\phi$ source to their endpoints, it is still possible to route the flow to sinks $v \in V$ of capacity $\mathbf{deg}_G(v)$. If this is the case the graph is guaranteed to be a $\phi/10$-expander. Otherwise, a sparse cut $A$ is detected by the algorithm and thereafter the remaining graph $G[V \setminus A]$ is guaranteed to be a $\phi/10$-expander. On the right: Vertex sets $A_1, A_2$ and $A_3$ are yet to be pruned from the graph. Each of these maintains a flow certificate, such that adding them up ensures that the remaining graph is a $\tilde{\Omega}(\phi)$-expander.
  • Figure 2: $C$ is a $\gamma$-bottleneck cut with respect to $(A, B)$ if $\frac{8}{\phi}(\text{\#red} - \text{\#blue}) + \frac{8}{\phi}(\mathbf{deg}_{G[V \setminus A']} - \mathbf{deg}_{G[V \setminus A'] \setminus B})(C \setminus A') + \frac{i}{\lambda}\mathop{\mathrm{\mathbf{vol}}}\nolimits_G(C \setminus A') \leq \gamma$ for every set $A' \supseteq A$. Notice that $\frac{i}{\lambda}\mathop{\mathrm{\mathbf{vol}}}\nolimits_G(C \setminus A') = \tt(C \setminus A')$ and that $\frac{8}{\phi}(\mathbf{deg}_{G[V \setminus A']} - \mathbf{deg}_{G[V \setminus A'] \setminus B})(C \setminus A')$ corresponds to the amount of source added for endpoints of deleted edges in $B$ inside the set $C \setminus A'$. In the figure, the deleted edges are marked with $x$ at both endpoints. For each such endpoint, a deleted edge adds $\frac{8}{\phi}$ source.

Theorems & Definitions (58)

  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: saranurak2021expander flow instance
  • Claim 2.2
  • proof
  • Definition 3.1: Decremental Expander Graph
  • proof
  • Lemma 4.1: Expansion Certificate, See Proposition A.3 in saranurak2021expander
  • proof
  • Claim 4.2
  • ...and 48 more