Cut Query Reachability for DAGs with Subquadratic Queries

2026-07-29

AI Summary

This work addresses the single-source reachability problem in directed acyclic graphs (DAGs) under the cut-query model. To overcome the high query cost of traditional approaches that require full graph reconstruction, the authors propose a deterministic algorithm that achieves subquadratic query complexity—the first such result for DAGs in this model. By integrating topological ordering with cut queries, the method efficiently infers reachability relations without explicitly constructing the graph structure and naturally extends to single-source shortest paths. The algorithm requires only $O(n\sqrt{n \log n})$ queries, significantly improving upon the previous randomized graph-reconstruction approach with $O(n^2 / \log n)$ query complexity.

Abstract

In the cut-query model, we have access to a (directed) graph via an oracle and we can query the size of the (directed) cut of a given subset of the vertices. One of the most elementary tasks in this model is to decide if there is a path two fixed vertices $s$ and $t$. While many results are known for undirected graphs, much less in understood for directed graphs in the cut query model. Even for the basic task of $s$-$t$ reachability, the best known randomized algorithm, is to reconstruct the entire graph with a technique by Grebinski and Kucherov using $O(n^2 / \log n)$ queries [Grebinski and Kucherov, 2000]. We restrict our attention to directed acyclic graphs (DAGs) and obtain a deterministic single-source reachability algorithm using $O(n \sqrt{n \log n})$ queries. The result is based on a topological sort algorithm, and can also be adapted to compute single-source shortest paths in DAGs.

Problem

• cut query
• reachability
• directed acyclic graph
• query complexity

Innovation

• cut-query model
• directed acyclic graphs
• reachability
• topological sorting
• subquadratic queries