  The network reconfiguration problem seeks to find a rooted tree $T$ such that
the energy of the (unique) feasible electrical flow over $T$ is minimized. The
tree requirement on the support of the flow is motivated by operational
constraints in electricity distribution networks. The bulk of existing results
on convex optimization over vertices of polytopes and on the structure of
electrical flows do not easily give guarantees for this problem, while many
heuristic methods have been developed in the power systems community as early
as 1989. Our main contribution is to give the first provable approximation
guarantees for the network reconfiguration problem. We provide novel lower
bounds and corresponding approximation factors for various settings ranging
from $\min\{O(m-n), O(n)\}$ for general graphs, to $O(\sqrt{n})$ over grids
with uniform resistances on edges, and $O(1)$ for grids with uniform edge
resistances and demands. To obtain the result for general graphs, we propose a
new method for (approximate) spectral graph sparsification, which may be of
independent interest. Using insights from our theoretical results, we propose a
general heuristic for the network reconfiguration problem that is orders of
magnitude faster than existing methods in the literature, while obtaining
comparable performance.
