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 $Ω(\log n)$ 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 $(1+ε)$-approximate distance preservers [Hoppenworth, Xu, Xu, SODA'25] and single-source minimum cut [Cheung, Lau, Leung, SICOMP'13], and obtain simpler construction of $(n^{1/3},ε)$-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.

翻译：我们证明，任意拥有 $n$ 个顶点和 $m$ 条边的有向图 $G$，均存在一个具有 $m^{1+o(1)}$ 条边的有向无环图（DAG），称为 DAG 投影，它能够实现以下任意一种功能：要么在 $G$ 中，对所有顶点对 $(s,t)$ 之间的距离进行 $(1+1/\text{polylog} (n))$ 近似；要么在 $G$ 中，对所有顶点子集对 $(S,T)$ 之间的最大流进行 $n^{o(1)}$ 近似。此前类似的结果在距离问题上存在 $\Omega(\log n)$ 的近似因子 [Assadi, Hoppenworth, Wein, STOC'25][Filtser, SODA'26]，而对于最大流问题，此前尚无此类结果。我们的 DAG 投影可在 $m^{1+o(1)}$ 时间内构建。此外，它们允许近乎最优的并行构建，即工作量为 $m^{1+o(1)}$、深度为 $m^{o(1)}$ 的算法（假设存在针对 DAG 上的近似最短路径或最大流问题的此类并行算法），即使输入图 $G$ 并非 DAG。DAG 投影能够直接将 DAG 上的结果（通常更简单且更高效）迁移至有向图上。作为示例，我们改进了 $(1+ε)$-近似距离保持器 [Hoppenworth, Xu, Xu, SODA'25] 和单源最小割 [Cheung, Lau, Leung, SICOMP'13] 的最新研究成果，并获得了 $(n^{1/3},ε)$-跳集 [Kogan, Parter, SODA'22][Bernstein, Wein, SODA'23] 和组合最大流算法 [Bernstein, Blikstad, Saranurak, Tu, FOCS'24][Bernstein, Blikstad, Li, Saranurak, Tu, FOCS'25] 的更简单构建方法。最后，通过 DAG 投影，我们将精确单源最短路径（SSSP）和最大流问题在近乎最优并行算法方面的几个主要开放难题简化为更易处理的情形：(1) 从精确有向 SSSP 简化为精确无向 SSSP；(2) 从精确有向 SSSP 简化为 DAG 上的 $(1+1/\text{polylog}(n))$ 近似问题；(3) 从精确有向最大流简化为 DAG 上的 $n^{o(1)}$ 近似问题。