Eulerian Graph Sparsification by Effective Resistance Decomposition

TL;DR

This paper introduces efficient algorithms for Eulerian graph sparsification using effective resistance decomposition, leading to faster solutions for Eulerian Laplacian systems and improved sparsifier bounds, with extensions to spectral sketches.

Contribution

It presents a novel effective resistance decomposition approach for Eulerian graph sparsification, improving runtime and sparsifier quality over prior methods.

Findings

Achieves near-linear time algorithms for Eulerian Laplacian systems.

Provides improved sparsifier size bounds compared to previous work.

Extends techniques to spectral graph sketching with polylogarithmic time complexity.

Abstract

We provide an algorithm that, given an $n$-vertex $m$-edge Eulerian graph with polynomially bounded weights, computes an $\breve{O}(n\log^{2} n \cdot \varepsilon^{-2})$-edge $\varepsilon$-approximate Eulerian sparsifier with high probability in $\breve{O}(m\log^3 n)$ time (where $\breve{O}(\cdot)$ hides $\text{polyloglog}(n)$ factors). Due to a reduction from [Peng-Song, STOC '22], this yields an $\breve{O}(m\log^3 n + n\log^6 n)$-time algorithm for solving $n$-vertex $m$-edge Eulerian Laplacian systems with polynomially-bounded weights with high probability, improving upon the previous state-of-the-art runtime of $\Omega(m\log^8 n + n\log^{23} n)$. We also give a polynomial-time algorithm that computes $O(\min(n\log n \cdot \varepsilon^{-2} + n\log^{5/3} n \cdot \varepsilon^{-4/3}, n\log^{3/2} n \cdot \varepsilon^{-2}))$-edge sparsifiers, improving the best such sparsity bound of $O(n\log^2 n \cdot \varepsilon^{-2} + n\log^{8/3} n \cdot \varepsilon^{-4/3})$ [Sachdeva-Thudi-Zhao, ICALP '24]. Finally, we show that our techniques extend to yield the first $O(m\cdot\text{polylog}(n))$ time algorithm for computing $O(n\varepsilon^{-1}\cdot\text{polylog}(n))$-edge graphical spectral sketches, as well as a natural Eulerian generalization we introduce. In contrast to prior Eulerian graph sparsification algorithms which used either short cycle or expander decompositions, our algorithms use a simple efficient effective resistance decomposition scheme we introduce. Our algorithms apply a natural sampling scheme and electrical routing (to achieve degree balance) to such decompositions. Our analysis leverages new asymmetric variance bounds specialized to Eulerian Laplacians and tools from discrepancy theory.