I am given a edge-weighted (multi)graph $G$ and two of its vertices, $u, v\in V(G)$. I want to find two edge-disjoint paths that connects $u$ and $v$ while minimizing the sum of the lengths of the paths.
what is the complexity of this problem ?
(If can reduce Hamiltonian Cycle to it if we ask for vertex-disjoint paths and allow negatives weights)
Anything known about approximation algorithms in general ? in planar graphs ?
(if we can cheat by an edge twice and paying it twice, then finding a shortest path gives a $2$-approximation.)
More generally, what if now I ask for $k$ edge-disjoints paths between from $u$ to $v$ ?
rq: this problem looks somehow related to subset TSP