[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83273-en":3,"doc-seo-83273-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83273,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Approximability of Electrical Distribution Network Reconfiguration for General Graphs","Distribution Network Reconfiguration (DNR) seeks a radial spanning tree of a power grid graph that minimizes total resistive loss, where each line loss equals its resistance times the squared line current. The work studies approximation algorithms and proves an n-approximation, alongside strong hardness: no n1−ε-approximation is possible for related planar Balanced Connected Partition unless P=NP. For many-source settings, results extend to k-DNR, including an Ω(log2 n) lower bound for 2-DNR, and APX-hardness plus an O(√n)-approximation for 1-DNR with uniform resistances.","arXiv :2607 .07600v 1 [ cs .DS] 8 Jul 2026  \nApproximability of Electrical Distribution Network Reconfiguration for General Graphs  \nChristian Wallisch 1,2 , Andrea Benigni 1,3,4 , Carsten Hartmann 1,3 , and  \nLeon Kellerhals5  \n1 Energy Systems Engineering (ICE-1), Institute of Climate and Energy Systems,  \nForschungszentrum Jülich, Germany  \n2 Hasso Plattner Institute, University of Potsdam, Germany  \n3 RWTH Aachen University, Germany  \n4 JARA-Energy, Jülich, Germany  \n5 TU Clausthal, Germany  \nAbstract  \nElectrical 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.  \nWe 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 −ε-approximation is possible even on planar graphs unless P = NP, for any ε > 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(log2 n) conditioned on P  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 O ( √n)-approximation for uniform line resistances, answering an open question by Gupta et al. [23] .  \n1 Introduction  \nWhen electric current flows through a power line, some power is converted into heat and thereby lost. This is captured by Joule’s law: The power loss on a line is the product of the line’s resistance and the square of the current flowing through it. This means that heavily loaded lines incur disproportionately large losses. It is therefore essential to avoid heavy loads on lines when distributing energy to households and businesses. While distribution networks (interpreted as undirected graphs whose edges are power lines) offer many connections to the consumers not least to offer redundancy [22], they are typically operated in a radial configuration: among all available lines, the active ones must form a spanning tree. For any such radial configuration, the production (negative demands) and  \nconsumption (positive demands) at the network vertices uniquely determine the flow on each active line, and hence the total power loss, which should be minimized. Formally, this is captured by the following optimization problem.  \nDistribution Network Reconfiguration (DNR)  \nInput: A connected undirected graph G = (V, E), a demand function  \nd: V → R satisfying Px∈V d (x) = 0, and a resistance function  \nr: E → R≥0 .  \nSolution: A spanning tree T of G.  \nObjective: Minimize L (T) := {x,y(T) r ({x, y}) v∈(Tx\\y) d (v) 2 .  \nHerein, for an edge {x, y}, the trees Tx\\y and Ty\\x are the unique components obtained by removing {x, y} from T , with x ∈ V(Tx\\y) and y ∈ V(Ty\\x) . Notably, it does not matter whether we pick Tx\\y or Ty\\x in LT: as G is connected and the sum of the demands is 0 , we have Pv∈V(Tx\\y) d (v) = −Pv∈V(Ty\\x) d (v) .  \nThe necessity of finding an energy-efficient radial configuration is evident. A recent CEER report [16] illustrates the scale: across 33 European countries, reported distribution losses ranged from 1 .95% to 22 .63%, with a median of 6 .31% .1 Moreover, temporal variability is becoming more pronounced as low-carbon technologies such as solar panels, electric vehicles, and heat pumps increase networ","cbCaipCPAI2dEPUm","https://ap.wps.com/l/cbCaipCPAI2dEPUm","pdf",705553,1,27,"English","en",105,"# Introduction\n## Problem formulation: Distribution Network Reconfiguration (DNR)\n## Motivation and related work\n## Main contributions","[{\"question\":\"What is the objective of Distribution Network Reconfiguration (DNR)?\",\"answer\":\"Given a graph of power lines with demands and resistances, DNR selects a spanning tree that minimizes total resistive power loss, where each edge’s loss is its resistance times the squared current induced by the tree.\"},{\"question\":\"What approximation guarantee is given by the paper for DNR?\",\"answer\":\"The paper provides an n-approximation algorithm for the studied DNR problem and analyzes when stronger approximation factors are impossible under standard complexity assumptions.\"},{\"question\":\"What hardness results does the paper establish?\",\"answer\":\"It shows that no n^{1−ε}-approximation is achievable even on planar graphs for a related planar Balanced Connected Partition variant unless P=NP, and it further derives conditioned lower bounds for k-DNR 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is the objective of Distribution Network Reconfiguration (DNR)?","Question",{"text":75,"@type":76},"Given a graph of power lines with demands and resistances, DNR selects a spanning tree that minimizes total resistive power loss, where each edge’s loss is its resistance times the squared current induced by the tree.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What approximation guarantee is given by the paper for DNR?",{"text":80,"@type":76},"The paper provides an n-approximation algorithm for the studied DNR problem and analyzes when stronger approximation factors are impossible under standard complexity assumptions.",{"name":82,"@type":73,"acceptedAnswer":83},"What hardness results does the paper establish?",{"text":84,"@type":76},"It shows that no n^{1−ε}-approximation is achievable even on planar graphs for a related planar Balanced Connected Partition variant unless P=NP, and it further derives conditioned lower bounds for k-DNR 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