Laying paths on a network using minimum number of links/edges
The problem you are asking is called capacitated network design. Typically each edge e also has a cost/weight w(e) and you want to minimize the weight/cost of the chosen edges such that the resulting graph supports the demands given by $T_{ij}$. The weighted version can be reduced to the unweighted version by replacing an edge of weight w(e) by a path of w(e) edges (this reduction is pseudo-polynomial). The problem is NP-Hard even when $T_{ij}$ is non-zero for a single pair and even here we don't quite understand the approximability. 

See the following recent papers for more on this:


Minimum-Cost Network Design with (Dis)economies of Scale. IEEE FOCS 2010.
Approximability of Capacitated Network Design. IPCO 2011.