Reachability in Directed Acyclic Graphs with Near-Linear Cut Queries

2026-07-23

AI Summary

This work addresses the long-standing challenge of achieving subquadratic query complexity for fundamental problems such as reachability in directed graphs under the cut-query model. Focusing on directed acyclic graphs (DAGs), the authors combine divide-and-conquer techniques with structural properties of DAGs to design the first algorithm with nearly linear query complexity. Specifically, single-source reachability and topological ordering can be determined using $O(n \log^3 n)$ cut queries. The approach is further extended to detect cycles in general directed graphs within the same asymptotic query bound. This result breaks a longstanding efficiency barrier in the cut-query model for directed graphs, marking a significant theoretical advance in the field.

Abstract

In the cut-query model, an algorithm is given access to a graph $G = (V, E)$ \emph{only} via cut queries. This model has seen significant attention in the undirected graph setting, with works establishing $O(n)$ cut query algorithms for computing the global minimum cut, $\widetilde{O}(n^{3/2})$ cut query algorithms for all pairs minimum cut, and many more. However, despite this vast array of progress in designing sub-quadratic query algorithms for computing properties of undirected graphs, there has been \emph{no} progress in designing such algorithms in directed graphs. Indeed, even for basic problems like whether a vertex $t$ is reachable from a vertex $s$, the cut query complexity is only known to be bounded in the interval $[Ω(n), O(n^2 / \log n)]$. In this work, we begin a systematic study of these basic problems in directed \emph{acyclic} graphs (DAGs). In this setting, we show that reachability from a single vertex and even topological sorting are both computable in $O(n \log^3 n)$ many cut queries. As a consequence, we also obtain an algorithm which, for any \emph{arbitrary} directed graph $G$, uses only $O(n \log^3 n)$ cut queries and determines whether $G$ contains a cycle.

Problem

• reachability
• directed acyclic graphs
• cut queries
• query complexity
• topological sorting

Innovation

• cut queries
• directed acyclic graphs
• reachability
• topological sorting
• cycle detection