Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs

TL;DR

This paper presents a dynamic algorithm for maintaining approximate all-pairs effective resistances in separable graphs with efficient update and query times, and proves complexity lower bounds based on the OMv conjecture.

Contribution

It introduces a fully dynamic algorithm for approximate effective resistances in separable graphs and develops a dynamic approximate Schur complement method, with complexity bounds and hardness results.

Findings

Achieves $ ilde{O}(rac{ oot{n}}{ ext{epsilon}^2})$ update and query time for separable graphs.

Develops a dynamic approximate Schur complement preserving pairwise resistances.

Proves lower bounds based on the OMv conjecture for both separable and general graphs.

Abstract

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an $n^{c}$-separator theorem for some $c<1$. We give a fully dynamic algorithm that maintains $(1+\varepsilon)$-approximations of the all-pairs effective resistances of an $n$-vertex graph $G$ undergoing edge insertions and deletions with $\tilde{O}(\sqrt{n}/\varepsilon^2)$ worst-case update time and $\tilde{O}(\sqrt{n}/\varepsilon^2)$ worst-case query time, if $G$ is guaranteed to be $\sqrt{n}$-separable (i.e., it is taken from a class satisfying a $\sqrt{n}$-separator theorem) and its separator can be computed in $\tilde{O}(n)$ time. Our algorithm is built upon a dynamic algorithm for maintaining \emph{approximate Schur complement} that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest. We complement our result by proving that for any two fixed vertices $s$ and $t$, no incremental or decremental algorithm can maintain the $s-t$ effective resistance for $\sqrt{n}$-separable graphs with worst-case update time $O(n^{1/2-\delta})$ and query time $O(n^{1-\delta})$ for any $\delta>0$, unless the Online Matrix Vector Multiplication (OMv) conjecture is false. We further show that for \emph{general} graphs, no incremental or decremental algorithm can maintain the $s-t$ effective resistance problem with worst-case update time $O(n^{1-\delta})$ and query-time $O(n^{2-\delta})$ for any $\delta >0$, unless the OMv conjecture is false.