Spectral Sparsification via Bounded-Independence Sampling

Abstract

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\le \log n$, and an error parameter $\epsilon >0$, our algorithm runs in space $\tilde{O}(k\log (N\cdot w_{\max }/w_{\min }))$ where $w_{\max }$ and $w_{\min }$ are the maximum and minimum edge weights in $G$, and produces a weighted graph $H$ with $\tilde{O}(n^{1+2/k}/{\epsilon }^2)$ edges that spectrally approximates $G$, in the sense of Spielmen and Teng [ST04], up to an error of $\epsilon$. Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava's effective resistance based edge sampling algorithm [SS08] and uses results from recent work on space-bounded Laplacian solvers [MRSV17]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by $k$ above, and the resulting sparsity that can be achieved.

37 pages