Loss Minimization for Electrical Flows over Spanning Trees on Grids

TL;DR

This paper proves the NP-hardness of the electrical flow minimization problem on grids and improves the approximation guarantee of the existing Min-Min algorithm using a novel convex optimization analysis.

Contribution

It establishes NP-hardness for grid cases and refines the Min-Min algorithm's approximation factor through a new convex optimization approach.

Findings

NP-hardness proven for grid-based electrical flow minimization.

Enhanced approximation guarantee for the Min-Min algorithm.

Novel analysis using convex optimization over a polytope.

Abstract

We study the electrical distribution network reconfiguration problem, defined as follows. We are given an undirected graph with a root vertex, demand at each non-root vertex, and resistance on each edge. Then, we want to find a spanning tree of the graph that specifies the routing of power from the root to each vertex so that all the demands are satisfied and the energy loss is minimized. This problem is known to be NP-hard in general. When restricted to grids with uniform resistance and the root located at a corner, Gupta, Khodabaksh, Mortagy and Nikolova [Mathematical Programming 2022] invented the so-called Min-Min algorithm whose approximation factor is theoretically guaranteed. Our contributions are twofold. First, we prove that the problem is NP-hard even for grids; this resolves the open problem posed by Gupta et al. Second, we give a refined analysis of the Min-Min algorithm and improve its approximation factor under the same setup. In the analysis, we formulate the problem of giving an upper bound for the approximation factor as a non-linear optimization problem that maximizes a convex function over a polytope, which is less commonly employed in the analysis of approximation algorithms than linear optimization problems.