Assigning edge weights under shortest path constraints
Yes.

Consider an exact assignment as described in your second paragraph.

If any edge has a weight $w_{a,b} &gt; c_{a,b}$ there must be an indirect path with total path weight $c_{a,b}$, This means the direct edge $ab$ will not be part of any minimal path.  We can lower the weight to $w_{a,b} = c_{a,b}$ without changing this.

Therefore, if an exact assignment exists, $w_{a,b} = c_{a,b}$ (for all a,b) will be an exact assignment.

It is clear that this is the unique assignment with minimal total weight.