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DAG Projections: Reducing Distance and Flow Problems to DAGs

Bernhard Haeupler, Yonggang Jiang, Thatchaphol Saranurak

Abstract

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.

DAG Projections: Reducing Distance and Flow Problems to DAGs

Abstract

We show that every directed graph with vertices and edges admits a directed acyclic graph (DAG) with edges, called a DAG projection, that can either -approximate distances between all pairs of vertices in , or -approximate maximum flow between all pairs of vertex subsets in . Previous similar results suffer a 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 -time constructions. Further, they admit almost-optimal parallel constructions, i.e., algorithms with work and depth, assuming the ones for approximate shortest path or maximum flow on DAGs, even when the input 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 -approximate distance preservers [Hoppenworth, Xu, Xu, SODA'25] and single-source minimum cut [Cheung, Lau, Leung, SICOMP'13], and obtain simpler construction of -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 -approximation on DAGs, and (3) From exact directed maximum flow to -approximation on DAGs.

Paper Structure

This paper contains 99 sections, 44 theorems, 207 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Approximating Distances.
  3. Approximating Maximum Flow.
  4. Our Structural Results
  5. $(1+\epsilon)$-Distance-Preserving DAG.
  6. $n^{o(1)}$-Congestion-Preserving DAG.
  7. Almost-Optimal Size.
  8. Parallel Construction: Reductions to Approximations on DAGs.
  9. New Landscape of Parallel Shortest Paths and Maximum Flow
  10. Transferring Results from DAGs to Directed Graphs
  11. Distance preservers: improved.
  12. Hop-set: simplified.
  13. Potential applications.
  14. Combinatorial Max flow: simplified.
  15. Bounded Single-Source Max flow: improved.
  16. ...and 84 more sections

Key Result

Theorem 1.1

For any directed graph $G$ with edge weights from $\{0,1,2,\dots,\mathrm{poly}(n)\}$ and $\epsilon\ge1/\mathrm{polylog}(n)$, there exists a DAG projection $G'$ to $G$ of size $|E(G')|=m^{1+o(1)}$ and width $n^{o(1)}$ such that, for every $s,t\in V(G)$, Moreover, there is a randomized algorithm for computing $G'$ in $m^{1+o(1)}$ time. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Old and new landscapes of parallel SSSP and maximum flow algorithms. The green area highlights the settings where near-optimal parallel algorithms are known. The red area highlights the settings as hard as the exact directed setting. The solid arrows represent non-trivial efficient parallel reductions, while the dotted arrows represent trivial ones.

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Efficient parallel reductions
  • Corollary 1.1
  • Theorem 1.4
  • lemma 2.1: BringmannCF23BernsteinNW25
  • definition 2.2: Terminal Expanding
  • definition 2.3: Expander Hierarchy
  • theorem 3.1: Section 8 of agarwal2024parallel
  • definition 4.1: Graph Projections
  • ...and 72 more