Any directed graph admits a sparse DAG projection of size m^{1+o(1)} that (1+1/polylog n)-approximates all-pairs distances or n^{o(1)}-approximates subset maximum flows.
Polylog-time and near-linear work approximation scheme for undirected shortest paths
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DAG Projections: Reducing Distance and Flow Problems to DAGs
Any directed graph admits a sparse DAG projection of size m^{1+o(1)} that (1+1/polylog n)-approximates all-pairs distances or n^{o(1)}-approximates subset maximum flows.