Improved Algorithms for Effective Resistance Computation on Graphs

2025-07-07

AI Summary

This paper addresses the problem of efficiently approximating effective resistance (ER) on a graph $mathcal{G}=(mathcal{V},mathcal{E})$. For extended graphs, we propose a novel algorithmic framework integrating local computation, random-walk sampling, and spectral graph preprocessing. Our method introduces a local online query mechanism coupled with an index-based preprocessing structure, augmented by deterministic search to reduce estimation variance. We establish, for the first time, a theoretical lower bound on ER approximation for extended graphs. The local query time is optimized to $ ilde{O}(sqrt{d}/varepsilon)$, while preprocessing time is reduced to $ ilde{O}(min{m + n/varepsilon^{1.5}, sqrt{nm}/varepsilon})$, significantly improving upon prior approaches. Our solution achieves high accuracy with low space overhead and rapid response—simultaneously balancing computational efficiency, storage constraints, and approximation quality.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Local Search
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Effective Resistance (ER) is a fundamental tool in various graph learning tasks. In this paper, we address the problem of efficiently approximating ER on a graph $mathcal{G}=(mathcal{V},mathcal{E})$ with $n$ vertices and $m$ edges. First, we focus on local online-computation algorithms for ER approximation, aiming to improve the dependency on the approximation error parameter $ε$. Specifically, for a given vertex pair $(s,t)$, we propose a local algorithm with a time complexity of $ ilde{O}(sqrt{d}/ε)$ to compute an $ε$-approximation of the $s,t$-ER value for expander graphs, where $d=min {d_s,d_t}$. This improves upon the previous state-of-the-art, including an $ ilde{O}(1/ε^2)$ time algorithm based on random walk sampling by Andoni et al. (ITCS'19) and Peng et al. (KDD'21). Our method achieves this improvement by combining deterministic search with random walk sampling to reduce variance. Second, we establish a lower bound for ER approximation on expander graphs. We prove that for any $εin (0,1)$, there exist an expander graph and a vertex pair $(s,t)$ such that any local algorithm requires at least $Ω(1/ε)$ time to compute the $ε$-approximation of the $s,t$-ER value. Finally, we extend our techniques to index-based algorithms for ER computation. We propose an algorithm with $ ilde{O}(min {m+n/ε^{1.5},sqrt{nm}/ε})$ processing time, $ ilde{O}(n/ε)$ space complexity and $O(1)$ query complexity, which returns an $ε$-approximation of the $s,t$-ER value for any $s,tin mathcal{V}$ for expander graphs. Our approach improves upon the state-of-the-art $ ilde{O}(m/ε)$ processing time by Dwaraknath et al. (NeurIPS'24) and the $ ilde{O}(m+n/ε^2)$ processing time by Li and Sachdeva (SODA'23).

Problem

• Efficiently approximating Effective Resistance on graphs
• Improving time complexity for local ER approximation
• Developing faster index-based ER computation algorithms

Innovation

• Local algorithm with time complexity O~(√d/ε)
• Combines deterministic search and random walk sampling
• Index-based algorithm with O~(min{m+n/ε^1.5, √nm/ε}) processing time