Reachability in Directed Acyclic Graphs with Near-Linear Cut Queries

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, $\tilde{O}(n^{3/2})$ 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 $[\hat{I}\text{c◯}(n),O(n^2/\log n)]$. 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.