Given a flow network with a source $$$s$$$ and sink $$$t$$$ (each edge has a capacity $$$c_i$$$ and flow $$$f_i$$$). You need to find decomposition into $$$s-t$$$ paths, where each path has a flow value $$$f_p$$$, such that it results in this flow network. Also, not all paths are correct, and we only need consider "correct" paths (we can understand is path correct or not while traversing the network). It is guarantee what this decomposition exists.

First of all, how I found a flow network? I used linear programming to solve this task. I have some inequalities for flows and for paths, and now I want to restore the answer as a decompisition of $$$s-t$$$ paths.

Is there exists a polynomial algorithm to solve this? The number of paths doesn't matter.