  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 $(1+\epsilon)$-resistance sparsifier of size $\tilde
O(n/\epsilon)$, and conjecture this bound holds for all graphs on $n$ nodes. In
comparison, spectral sparsification is a strictly stronger notion and requires
$\Omega(n/\epsilon^2)$ 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.
