Towards Optimal Effective Resistance Estimation

Rajat Vadiraj Dwaraknath, Ishani Karmarkar, Aaron Sidford

Keywords: effective resistances, spectral sketch, fine-grained complexity, triangle detection, numerical linear algebra

TL;DR: We provide improved algorithms and conditional lower bounds for the problem of estimating effective resistances between nodes in an undirected graph, and apply the techniques to more general problems involving general positive semidefinite matrices.

Abstract:

We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high probability, an $\widetilde{O}(n\epsilon^{-1})$-bit sketch from which the effective resistance between any pair of nodes can be estimated, to $(1 \pm \epsilon)$-multiplicative accuracy, in $\widetilde{O}(1)$-time. Consequently, we obtain an $\widetilde{O}(m\epsilon^{-1})$-time algorithm for estimating the effective resistance of all edges in such graphs, improving (for sparse graphs) on the previous fastest runtimes of $\widetilde{O}(m\epsilon^{-3/2})$ [Chu et. al. 2018] and $\widetilde{O}(n^2\epsilon^{-1})$ [Jambulapati, Sidford, 2018] for general graphs and $\widetilde{O}(m + n\epsilon^{-2})$ for expanders [Li, Sachdeva 2022].
We complement this result by showing a conditional lower bound that a broad set of algorithms for computing such estimates of the effective resistances between all pairs of nodes require $\widetilde{\Omega}(n^2 \epsilon^{-1/2})$-time, improving upon the previous best such lower bound of $\widetilde{\Omega}(n^2 \epsilon^{-1/13})$ [Musco et. al. 2017]. Further, we leverage the tools underlying these results to obtain improved algorithms and conditional hardness for more general problems of sketching the pseudoinverse of positive semidefinite matrices and estimating functions of their eigenvalues.

Submission Number: 10879