Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

In this paper we provide an O(mloglog^O(1)nlog (1/ϵ))-expected time algorithm for solving Laplacian systems on n-node m-edge graphs, improving upon the previous best expected runtime of \(O(m \sqrt {\log n} \mathrm{log log}^{O(1)} n \log (1/\epsilon)) \) achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of low spectral stretch graph approximations with improved stretch and sparsity bounds. As motivation for this work, we show that for every set of vectors in \(\mathbb {R}^d \) (not just those induced by graphs) and all integer k > 1 there exist an ultra-sparsifier with d − 1 + O(d/k) re-weighted vectors of relative condition number at most k². For small k, this improves upon the previous best known multiplicative factor of \(k \cdot \tilde{O}(\log d) \), which is only known for the graph case. Additionally, in the graph case we employ our low-stretch subgraph construction to obtain n − 1 + O(n/k)-edge ultrasparsifiers of relative condition number k^{1 + o(1)} for k = ω(log ^δn) for any δ > 0: this improves upon the previous work for k = o(exp (log ^{1/2 − δ}n)).