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 transcript assembly. We prove the problem is in FPT when parameterized by the number of paths by giving a practical linear fpt algorithm. Further, we implement and engineer a Flow Decomposition solver based on this algorithm, and evaluate its performance on RNA-sequence data. Crucially, our solver finds exact solutions while achieving runtimes competitive with a state-of-the-art heuristic. Finally, we contextualize our design choices with two hardness results related to preprocessing and weight recovery. Specifically, k-Flow Decomposition does not admit polynomial kernels under standard complexity assumptions, and the related problem of assigning (known) weights to a given set of paths is NP-hard.

We analyze local properties of sparse Erdős-Rényi graphs, where d(n)/n is the edge probability. In particular we study the behavior of very short paths. For d(n) = n o(1) we show that G(n, d(n)/n) has asymptotically almost surely (a.a.s.) bounded local treewidth and therefore is a.a.s. nowhere dense. We also discover a new and simpler proof that G(n, d/n) has a.a.s. bounded expansion for constant d. The local structure of sparse Erdős-Rényi graphs is very special: The r-neighborhood of a vertex is a tree with some additional edges, where the probability that there are m additional edges decreases with m. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally, experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices.

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