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Minimum flow decomposition (MFD) -- the problem of finding a minimum set ofweighted source-to-sink paths that perfectly decomposes a flow -- is aclassical problem in Computer Science, and variants of it are powerful modelsin different fields such as Bioinformatics and Transportation. Even on acyclicgraphs, the problem is NP-hard, and most practical solutions have been viaheuristics or approximations. While there is an extensive body of research onacyclic graphs, currently, there is no \emph{exact} solution on graphs withcycles. In this paper, we present the first ILP formulation for three naturalvariants of the MFD problem in graphs with cycles, asking for a decompositionconsisting only of weighted source-to-sink paths or cycles, trails, and walks,respectively. On three datasets of increasing levels of complexity from bothBioinformatics and Transportation, our approaches solve any instance in under10 minutes. Our implementations are freely available atgithub.com/algbio/MFD-ILP.

Minimum Flow Decomposition in Graphs with Cycles using Integer Linear Programming