Aaron Bernstein, Henry Fleischmann, Bernhard Haeupler, Gary Hoppenworth, Thatchaphol Saranurak
The Steiner shortcut problem asks whether every directed graph $G=(V,E)$ with $n$ vertices and $m$ edges admits an augmentation $G_{+}=(V\cup V_{+},E\cup E_{+})$ with $m^{1+o(1)}$ edges such that $G_+$ has the same reachability structure as $G$ (i.e. for all $s,t\in V$, $s$ reaches $t$ in $G$ iff $s$ reaches $t$ in $G_+$), and the diameter of $G_{+}$ is $m^{o(1)}$. This is a core combinatorial question underlying many reachability algorithms in the dynamic, streaming, and CONGEST models, and all parallel algorithms with almost-linear work and sublinear depth. A sequence of works ruled out such a shortcut $G_{+}$ in the non-Steiner setting where $V_{+}=\emptyset$. However, as recently shown by Bernstein et al. [STOC '26], all known non-Steiner lower-bound constructions fail once Steiner vertices $V_{+}$ are allowed. This paper rules out almost-optimal Steiner shortcuts: we show an infinite family of graphs $G=(V,E)$ with the property that, for any $\varepsilon>0$, a Steiner shortcut $G_{+}=(V\cup V_{+},E\cup E_{+})$ with $m^{1+\varepsilon}$ edges must have diameter at least $n^{1/56-O(\varepsilon)}$. Our construction strengthens the standard non-Steiner lower-bound framework via an additional structure that allows Steiner vertices to be replaced by original vertices; we can then reduce back to the standard arguments without Steiner vertices. This new structure is achieved via a new general strategy based on test functions, which is quite flexible and can potentially be extended to applications beyond our work. We also generalize our lower bound to exact Steiner hopsets in undirected graphs and to depth reduction for monotone circuits. The latter significantly strengthens the classical lower bound based on Sylvester matrices.
A Simple Subquadratic Cut Query Protocol for Directed Reachability
Henry Fleischmann, Sagnik Mukhopadhyay, Swagato Sanyal
AI 速览：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。
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.