Eulerian Graph Sparsification by Effective Resistance Decomposition
Arun Jambulapati, Sushant Sachdeva, Aaron Sidford, Kevin Tian, Yibin Zhao
TL;DR
This work advances directed graph sparsification by introducing an effective resistance (ER) decomposition framework that preserves degree balance while enabling near-independent edge-signing analysis. By combining random signing with electrical routing and a novel asymmetric variance bound parameterized by ER diameter, the authors achieve Eulerian sparsifiers with near-optimal sparsity and a nearly-linear-time construction. This leads to faster Eulerian Laplacian solvers and broader directed-graph primitives, leveraging discrepancy-theoretic ideas in a directed setting. The framework also yields graphical spectral sketches for Eulerian graphs, unifying sparsification, sketching, and inverse-Laplacian-like approximations under a coherent, efficient scheme. Overall, the results close gaps between sparsity and runtime in Eulerian sparsification and push forward practical directed-graph algorithms.
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 $Ω(m\log^8 n + n\log^{23} n)$. 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.