  We consider network routing under random link failures with a desired final
distribution. We provide a mathematical formulation of a relaxed transport
problem where the final distribution only needs to be close to the desired one.
The problem is a maximum entropy problem for path distributions with an extra
terminal cost. We show that the unique solution may be obtained solving a
generalized Schroedinger system. An iterative algorithm to compute the solution
is provided. It contracts the Hilbert metric with contraction ratio less than
1/2 leading to extremely fast convergence.
