Electrical flows over spanning trees

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