Approximability of Electrical Distribution Network Reconfiguration for General Graphs

2026-07-08

AI Summary

This study addresses the Distribution Network Reconfiguration (DNR) problem, which seeks to minimize resistive power losses under radial topology constraints. Combining combinatorial optimization, approximation algorithm design, and complexity theory, the work establishes several key results for both single- and multi-feeder settings. It proves for the first time that DNR admits no $n^{1-\varepsilon}$-approximation algorithm even on planar graphs unless P = NP. For the single-feeder case, the paper presents an $O(\sqrt{n})$-approximation algorithm and establishes its APX-hardness, resolving an open question posed by Gupta et al. In the two-feeder setting, it derives an $\Omega(\log^2 n)$ lower bound on approximability and provides a general $n$-approximation algorithm applicable to arbitrary numbers of feeders.

Abstract

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 $Ω(\log^2 n)$ 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].

Problem

• Distribution Network Reconfiguration
• Power Loss Minimization
• Spanning Tree
• Resistive Losses
• Approximability

Innovation

• Distribution Network Reconfiguration
• Approximation Algorithms
• NP-hardness
• Confluent Flow
• Power Loss Minimization