有向无环图中近线性割查询的可达性
Reachability in Directed Acyclic Graphs with Near-Linear Cut Queries
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中文总结 AI 辅助
研究有向无环图基本问题，通过\(O(n \log^3 n)\)次割查询可计算单顶点可达性与拓扑排序，由此得到判断任意有向图是否含环的算法，填补有向图割查询算法设计的空白。
中文摘要 AI 辅助
在割查询模型中，算法只能通过割查询访问图\(G = (V, E)\)。该模型在无向图设置中备受关注，已有计算全局最小割的\(O(n)\)割查询算法等。但在有向图中设计此类算法尚无进展，如顶点可达性问题的割查询复杂度在\([\Omega(n), O(n^2 / \log n)]\)。本文对有向无环图中的这些基本问题进行系统研究，证明单顶点可达性和拓扑排序可在\(O(n \log^3 n)\)割查询内完成，还得到判断任意有向图是否含环的算法。
英文摘要
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.
