Motivated by transcript assembly and multiple genome assembly problems, in this paper, we study the following minimum path flow decomposition problem: given a directed acyclic graph G=(V, E) with source s and sink t and a flow f, compute a set of s-t paths P and assign weight w(p) for p ∈ P such that f(e) = Σp∈P:e∈pw(p), ∀e ∈ E, and |P| is minimized. We propose an efficient pseudo-polynomial-time heuristic for this problem based on novel insights. Our heuristic gives a framework that consists of several components, providing a roadmap for continuing development of better heuristics. Through experimental studies on both simulated and transcript assembly instances, we show that our algorithm significantly improves the previous state-of-the-art algorithm. Implementation of our algorithm is available at https://github.com/Kingsford-Group/catfish.