  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.
