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 \(\min \{{\mathcal {O}}(m-n), {\mathcal {O}}(n)\}\) for general graphs, to \({\mathcal {O}}(\sqrt{n})\) over grids with uniform resistances on edges, and \({\mathcal {O}}(1)\) 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.

的网络重新配置问题寻求发现有根树Ť使得（唯一的）是可行的电气过流的能量Ť被最小化。配电网络中的操作约束促使对流的支持树要求。关于多边形顶点的凸优化和电流结构的大量现有结果并不容易为这个问题提供保证，而早在1989年电力系统界就已经开发了许多启发式方法。我们的主要贡献是：为网络重新配置问题提供第一个可证明的近似保证。我们为各种设置提供了新颖的下界和相应的近似因子，范围从\（\ min \ {{\ mathcal {O}}（mn），{\ mathcal {O}}（n）\} \）到普通图，到\（{\ mathcal {O}}（\ sqrt {n }）\）在边缘上具有均匀电阻的网格上，而\（{\ mathcal {O}}（1）\）在边缘上具有均匀电阻和要求的网格上。为了获得通用图的结果，我们提出了一种（近似）频谱图稀疏化的新方法，该方法可能具有独立的意义。利用我们的理论结果得出的见解，我们提出了一种针对网络重新配置问题的一般启发式方法，该方法比文献中的现有方法快了几个数量级，同时获得了相当的性能。