Maximum fractional packing of spanning trees. Given a graph $G$, let $\{T_1,\ldots,T_k\}$ be a set of spanning trees with associated nonnegative weights $\{w_1,\ldots,w_k\}$ such that for every edge $e$, $\sum_{e\in T_i}w_i \leq 1$.  This is a fractional packing of spanning trees.

What is the maximum total weight of such a fractional packing, i.e. the maximum possible value of $\sum_{i=1}^kw_i$?

Can this be done in polynomial time?  If so, how?

(Note that the dual linear program asks for the minimum total edge weight such that every spanning tree receives weight at least one, so the dual is a minimum fractional edge cut problem.)
The problem gets very easy if you look at the feasible space, which is simply the spanning tree polytope, intersected with a linear number of half-spaces, one for each packing constraint. 

The spanning tree polytope has a separation oracle (check this for example), therefore since you only have an additional linear number of constraints, it carries over to yield a separation oracle for your problem. Using this oracle, you can optimize any linear functional of the polytope in polynomial time.

If I remember well, using the ellipsoid algorithm, you can get a running time of $\tilde{O}(m^2 \cdot T)$, where $T$ is the running time of the separation oracle (which in this case you can get from a min-cut algorithm). Probably not as good as other existing algorithms, but it gives you an idea of how you can quickly check that the problem is polynomial time solvable.