Approximability of Electrical Distribution Network Reconfiguration for General Graphs

Electrical distribution networks are regional, medium- and low-voltage power grids connecting energy sources to individual households and businesses with given power demands. While these networks contain redundant power lines for reliability, they are typically operated in a radial (spanning tree) configuration by opening and closing switches on the lines. The challenge is to find a spanning tree that minimizes the sum of the resistive power losses: The power loss of a line $e$ is its resistance $r(e)$ times the squared current $f(e)^2$ flowing across the line. We study approximation algorithms for this problem, known as Distribution Network Reconfiguration (DNR). We give an $n$-approximation algorithm and, via a new NP-hardness for planar Balanced Connected Partition with a fixed number of parts, show that no $n^{1-\varepsilon}$-approximation is possible even on planar graphs unless P $=$ NP, for any $\varepsilon>0$. Since the approximation hardness holds only if there are many sources, we focus on $k$-DNR with $k$ sources; this is motivated by traditional distribution networks, where oftentimes $k = 1$. For $2$-DNR, we give an approximation lower bound of $\Omega(\log^2 n)$ conditioned on P $\neq$ NP. For $1$-DNR, which is equivalent to finding an uncapacitated confluent flow minimizing the squared Euclidean norm, we prove APX-hardness and give an $\mathcal{O}(\sqrt{n})$-approximation for uniform line resistances, answering an open question by Gupta et al. [Math. Program. 2022].