DAG Projections: Reducing Distance and Flow Problems to DAGs

We show that every directed graph $G$ with $n$ vertices and $m$ edges admits a directed acyclic graph (DAG) with $m^{1+o(1)}$ edges, called a DAG projection, that can either $(1+1/\text{polylog} (n))$-approximate distances between all pairs of vertices $(s,t)$ in $G$, or $n^{o(1)}$-approximate maximum flow between all pairs of vertex subsets $(S,T)$ in $G$. Previous similar results suffer a $\Omega(\log n)$ approximation factor for distances [Assadi, Hoppenworth, Wein, STOC'25] [Filtser, SODA'26] and, for maximum flow, no prior result of this type is known. Our DAG projections admit $m^{1+o(1)}$-time constructions. Further, they admit almost-optimal parallel constructions, i.e., algorithms with $m^{1+o(1)}$ work and $m^{o(1)}$ depth, assuming the ones for approximate shortest path or maximum flow on DAGs, even when the input $G$ is not a DAG. DAG projections immediately transfer results on DAGs, usually simpler and more efficient, to directed graphs. As examples, we improve the state-of-the-art of $(1+\epsilon)$-approximate distance preservers [Hoppenworth, Xu, Xu, SODA'25] and single-source minimum cut [Cheung, Lau, Leung, SICOMP'13], and obtain simpler construction of $(n^{1/3},\epsilon)$-hop-set [Kogan, Parter, SODA'22] [Bernstein, Wein, SODA'23] and combinatorial max flow algorithms [Bernstein, Blikstad, Saranurak, Tu, FOCS'24] [Bernstein, Blikstad, Li, Saranurak, Tu, FOCS'25]. Finally, via DAG projections, we reduce major open problems on almost-optimal parallel algorithms for exact single-source shortest paths (SSSP) and maximum flow to easier settings: (1) From exact directed SSSP to exact undirected ones, (2) From exact directed SSSP to $(1+1/\text{polylog}(n))$-approximation on DAGs, and (3) From exact directed maximum flow to $n^{o(1)}$-approximation on DAGs.