Electrical Flows over Spanning Trees
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This is the first paper to give provable approximation guarantees for the network reconfiguration problem from power systems. The problem seeks to find a rooted tree such that the energy of the (unique) feasible electrical flow is minimized. The tree requirement is motivated by operational constraints in electricity distribution networks. The bulk of existing results on the structure of electrical flows, Laplacian solvers, bicriteria tree approximations, etc., do not easily give guarantees for this problem, while many heuristic methods have been used in the power systems community as early as 1989. Our main contribution is to advance the theory for network reconfiguration by providing novel lower bounds and corresponding approximation factors for various settings ranging from O(n) for general graphs, to O(√(n)) over grids with uniform resistances on edges, and O(1) for grids with uniform edge resistances and demands. We also provide a new method for (approximate) graph sparsification that maintains the original resistances of the edges.
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