Eulerian Graph Sparsification by Effective Resistance Decomposition

Abstract: We provide an algorithm that, given an $n$-vertex $m$-edge Eulerian graph with polynomially bounded weights, computes an $$\breve{O}(n\log^{2} n \cdot \varepsilon<sup>{-2})$$-edge $\varepsilon$-approximate Eulerian sparsifier with high probability in $\breve{O}(m\log³ n)$ time (where $\breve{O}(\cdot)$ hides $\text{polyloglog}(n)$ factors). Due to a reduction from [Peng-Song, STOC '22], this yields an $$\breve{O}(m\log³ n + n\log⁶ n)$$-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 $$\Omega(m\log⁸ n + n\log^{23} n)$$. We also give a polynomial-time algorithm that computes $$O(\min(n\log n \cdot \varepsilon^{-2} + n\log^{5/3} n \cdot \varepsilon^{-4/3}, n\log^{3/2} n \cdot \varepsilon<sup>{-2}))$$-edge sparsifiers, improving the best such sparsity bound of $$O(n\log² n \cdot \varepsilon^{-2} + n\log^{8/3} n \cdot \varepsilon<sup>{-4/3})$$ [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 $O(n\varepsilon<sup>{-1}\cdot\text{polylog}(n))$-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.