A practical fpt algorithm for Flow Decomposition and transcript assembly

The Flow Decomposition problem, which asks for the smallest set of weighted paths that “covers” a flow on a DAG, has recently been used as an important computational step in genetic assembly problems. We prove the problem is in FPT when parameterized by the number of paths, and we give a practical linear fpt algorithm. Combining this approach with algorithm engineering, we implement a Flow Decomposition solver and demonstrate its competitiveness with a state-of-the-art heuristic on RNA sequencing data. We contextualize our design choices with two hardness results related to preprocessing and weight recovery. First, the problem does not admit polynomial kernels under standard complexity assumptions. Second, the related problem of assigning weights to a given set of paths is NP-hard even when the weights are known.