I'm curious to find an algorithm that solves the following graph-theory problem.

Suppose I have a graph $G(V,E)$ with two disjoint sets of vertices, $V_a$ and $V_b$.

My goal is to find paths from every vertex in $V_a$ to every vertex in $V_b$ where the edges in these paths are minimally overlapping. Here we define two paths to be overlapping if they share a same edge. When we say minimally overlapping this can be quantified by measuring the weights of overlapping edges (e.g., two overlapping edges with total weight of 5 is better than one overlapping edge with weight of 10).

Does such an algorithm exist?