Is any related work to this m-trails problem ? Yesterday, I discussed with one of my EE friends. She asked me an interesting problem and I simplify it by ignoring the bandwidth cost and model as following:
Given a graph $G=(V,E)$ with its path set $P=\{P_1,P_2,\ldots, P_m\}$ where $P_i$ is a path 
between two points in $V$. A path $P_i$ is colored by red if only and only if one of it's edges is colored by red, i.e. $P^c=red$; otherwise $P_i$ is colored by blue, i.e. $P^c=blue$. Find a subset $P_s\subseteq P$ s.t. 1) If for every $j$ color all edges in $G$ by blue except only one $e_j\in E$ by red, 
there is a subset $P_s&#39; =\{P_1,P_2,\ldots,P_t\}$ in $P_s$ such that $f(P_1^c,\ldots,P_t^c)=e_j$ where f() is one-to-one mapping; 2) minimize the size of the result set $P_s$.

My questions are:
1) is there any similar work done in TCS?
2) is there any similar work done in graph theory?
3)2) is there any similar work done in networks?
3) any discussions about this problem are welcome.
What is $f$? I seems that you're asking for a one-to-one mapping between a power set over a power set of edges and the edges - which would surely be possible only if there's no more than one edge.