I am reading the Yoshua Bengio et al, Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation. It seems to me the objective of the paper is to generate the flow $F$ given the reward function $R(s)$ that satisfies the detailed balance constraint Equation (4). This can be easily solved by walking backwards from the terminal nodes and accumulating reward along the way. When the incident action edge is not unique the flow solution is not unique and in fact uncountably infinite number of them. The computational complexity of this algorithm is obviously linear with respect to the number of states.
What is the rationale for devising an objective function such as Equation (12) which is computationally complex to solve the problem?