DAG投影：将距离和流问题缩减到DAGs
DAG Projections: Reducing Distance and Flow Problems to DAGs
AI总结：
本文通过DAG投影将距离和流问题转化为DAGs，实现近似距离和最大流的高效算法，同时简化了相关问题的并行计算。
摘要：
我们证明了每个具有n个顶点和m条边的有向图G都允许一个具有m^{1+o(1)}条边的有向无环图（DAG），称为DAG投影，可以以(1+1/polylog(n))的精度近似所有顶点对的距离，或以n^{o(1)}的精度近似所有顶点子集对的最大流。先前类似的结果在距离问题上需要Ω(log n)的近似因子[Assadi, Hoppenworth, Wein, STOC'25] [Filtser, SODA'26]，而在最大流问题上尚无此类结果。我们的DAG投影具有m^{1+o(1)}的时间构造。此外，它们还允许几乎最优的并行构造，即具有m^{1+o(1)}工作量和m^{o(1)}深度的算法，假设存在近似最短路径或最大流的DAG算法，即使输入图G不是DAG。DAG投影立即将DAG上的结果转移到有向图上，通常更简单且更高效。例如，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]的状态-of-the-art，以及获得了( n^{1/3}, ε)-hop-set的更简单的构造[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]。最后，通过DAG投影，我们将主要的开放问题——几乎最优的并行算法对于精确的单源最短路径（SSSP）和最大流问题——简化到更简单的设置：(1) 从精确的有向SSSP到精确的无向SSSP；(2) 从精确的有向SSSP到DAG上的(1+1/polylog(n))近似；(3) 从精确的有向最大流到DAG上的n^{o(1)}近似。
英文摘要：
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.
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