Towards Resistance Sparsifiers

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+ϵ)$-resistance sparsifier of size $\tilde O(n/\epsilon)$,andconjecturethisboundholdsforallgraphson$n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $\mathrm{\Omega }(n\mathrm{/}{ϵ}^2)$ edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers.