Loss Minimization for Electrical Flows over Spanning Trees on Grids

TL;DR

This work investigates the EDNR problem on graphs by formalizing the loss of a spanning tree as $L(T) = \sum_{e \in F} r_e (\sum_{v \in Des_T(e)} d_v)^2$ and proving NP-hardness even for grid topologies. It advances the theoretical understanding by providing two hardness reductions (height-3 grids and binary demands) and by refining the Min-Min approximation: for grids with uniform demands and resistances and a corner root, the algorithm achieves a $(9/8+o(1))$-approximation, improving prior bounds of $2+o(1)$, with extensions to nonuniform demands via a factor of $\alpha^2$. The analysis introduces a non-linear convex-optimization formulation to bound the approximation factor, a less common technique in this area. Collectively, the results sharpen the complexity landscape for EDNR on grids and deliver a tighter, practically relevant approximation bound for a canonical grid setting, while leaving open the exact optimal-tree shape on such grids and suggesting avenues for further refinement of the analytical toolkit.

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.

Figures (7)

• Figure 1: An instance of the EDNR problem. The root is specified with $r$. Each non-root vertex is associated with its demand. Each edge is associated with its resistance. Red edges show a spanning tree $T$, and a red boxed number shows $\sum_{v \in \mathrm{Des}_{T}(e)}d_{v}$ for each edge $e$ of the tree $T$.
• Figure 2: The comparison of an optimal solution (left) and the output of the Min-Min algorithm (right) for the $7\times 7$ grid with uniform demand and resistance. The optimal loss is $3{,}242$ while the loss from the Min-Min algorithm is $3{,}246$.
• Figure 3: Proof of Theorem \ref{['thm:hard_constheight']}. Thick edges have resistance $1$ and thin edges have resistance $0$.
• Figure 4: Proof of Theorem \ref{['thm:hard_binarydemand']}. Thick edges have resistance $1$, thin edges have resistance $0$, and missing edges (white edges, invisible edges) have resistance $\infty$. White vertices have demand $1$ and non-root black vertices have demand $0$. The blue window shows a close-up image of the chain with $a_i$ vertices of demand $1$. The red window shows a close-up image of one of the resistance-$1$ edges.
• Figure 5: Construction of an optimal spanning tree in the proof of Theorem \ref{['thm:hard_binarydemand']}, shown by red edges. In the figure, several black vertices are not incident with any red edges, but they can be connected to the existing red tree only via black edges (i.e., edges of resistance $0$), which incurs no extra cost.

Theorems & Definitions (20)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Corollary 1
• Lemma 1
• proof
• Lemma 2
• proof