The Complexity of Network Design for s-t Eff ective Resistance

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 $G=(V,E)$ where each edge $e$ has a cost $c_e$ and a resistance $r_e$, two designated vertices $s,t \in V$, and a cost budget $k$. Our goal is to choose a subgraph to minimize the $s$-$t$ effective resistance, subject to the constraint that the total cost in the subgraph is at most $k$. This problem has applications in electrical network design and is an interpolation between the shortest path problem and the minimum cost flow problem.

We present algorithmic and hardness results for this problem. On the hardness side, we show that the problem is NP-hard by reducing the 3-dimensional matching problem to our problem. On the algorithmic side, we use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph. Finally, we propose a greedy algorithm for general graphs in which we add a path at each iteration and we conjecture that the algorithm is a $3.95$-approximation algorithm for the problem.