Electrical Flows over Spanning Trees

TL;DR

This paper introduces the first provable approximation guarantees for electrical network reconfiguration over spanning trees, providing bounds for different graph types and proposing a fast heuristic based on spectral sparsification.

Contribution

It offers novel approximation bounds for the network reconfiguration problem and introduces a new spectral graph sparsification method with practical heuristic algorithms.

Findings

Approximation factors range from O(m-n) to O(1) depending on graph type.

New spectral sparsification method developed for approximate solutions.

Heuristic algorithm achieves comparable results with significantly faster computation.

Abstract

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.