# Network design for s-t effective resistance

## TL;DR

This paper studies the problem of designing a subgraph with limited edges to minimize the effective resistance between two nodes, providing hardness results and approximation algorithms, including a PTAS for series-parallel graphs.

## Contribution

It introduces the first approximation algorithms and hardness results for the s-t effective resistance network design problem, bridging shortest path and flow problems.

## Key findings

- The problem is NP-hard and hard to approximate within a factor of two.
- A convex relaxation yields a constant factor approximation algorithm.
- A PTAS exists for series-parallel graphs, improving approximation ratios.

## Abstract

We consider a new problem of designing a network with small $s$-$t$ effective resistance. In this problem, we are given an undirected graph $G=(V,E)$, two designated vertices $s,t \in V$, and a budget $k$. The goal is to choose a subgraph of $G$ with at most $k$ edges to minimize the $s$-$t$ effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.