Electrical flows over spanning trees

• 成果类型: Article
• 署名作者: Gupta, Swati; Khodabakhsh, Ali; Mortagy, Hassan; Nikolova, Evdokia
• 署名单位: University System of Georgia; Georgia Institute of Technology; University of Texas System; University of Texas Austin
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01614-x
• 发表日期: 2022
• 页码: 479-519

摘要

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(n)for general graphs, to O(root 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 ofmagnitude faster than existingmethods in the literature, while obtaining comparable performance.