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新论文 · v1arXiv:2611.06906

一个有向可达性的简单次二次割查询协议

A Simple Subquadratic Cut Query Protocol for Directed Reachability

Henry Fleischmann, Sagnik Mukhopadhyay, Swagato Sanyal

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作者提出一个简单的随机化算法蓝图,通过有向割查询在真正次二次查询复杂度内判定有向图 s–t 可达性,给出两种实现分别达到 Õ(n^{9/5}) 与 Õ(n^{5/3}) 查询上界。

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We study $s$--$t$ reachability in an unweighted directed graph on $n$ vertices, accessed through a directed cut oracle: given $S\subseteq V$, the oracle returns the number of edges directed from $S$ to $V\setminus S$. Directed cut functions are submodular, and reachability asks whether the minimum $s$--$t$ cut has positive value. This connects the problem to the query complexity of submodular function minimisation. Whether or not reachability can be solved in truly subquadratic query complexity has been an outstanding open question for a decade. We give a simple algorithmic blueprint with two randomised implementations. The first uses $\widetilde O(n^{9/5})$ outgoing-cut queries; the second improves this bound to $\widetilde O(n^{5/3})$. The common idea is to construct an alternative orientation with the same underlying undirected graph and vertex-wise degree imbalances, thereby preserving reachability. We first approximate the target imbalances using an averaging and rounding procedure, then correct the remaining error with a flow. An earlier manuscript, made available in August 2026 in an online AI repository, reports an $\widetilde O(n^{3/2})$ query bound. Our contribution is a simple and comprehensible alternative approach that already achieves a polynomial saving over quadratic query complexity.