Given a network G(V,A,c) and a collection of origin-destination pairs with
prescribed values, the reverse shortest path problem is to modify the arc
length vector c as little as possible under some bound constraints such that
the shortest distance between each origin-destination pair is upper bounded
by the corresponding prescribed value. It is known that the reverse shortest
path problem is NP-hard even on trees when the arc length modifications are
measured by the weighted sum-type Hamming distance. In this paper, we
consider two special cases of this problem which are polynomially solvable.
The first is the case with uniform lengths. It is shown that this case
transforms to a minimum cost flow problem on an auxiliary network. An
efficient algorithm is also proposed for solving this case under the unit
sum-type Hamming distance. The second case considered is the problem without
bound constraints. It is shown that this case is reduced to a minimum cut
problem on a tree-like network. Therefore, both cases studied can be solved
in strongly polynomial time.