We study the space complexity of computing a sparse subgraph of a directed graph that certifies connectivity in the streaming and distributed models. Formally, for a directed graph $G=(V,A)$ and $k\in \mathbb{N}$, a $k$-node strong connectivity certificate is a subgraph $H=(V,A')\subseteq G$ such that for every pair of distinct nodes $s,t\in V$, the number of pairwise internally node-disjoint paths from $s$ to $t$ in $H$ is at least $k$ or the corresponding number in $G$. In light of the inherent hardness of directed connectivity problems, several prior work focused on restricted graph classes, showing that several problems that are hard in general become efficiently solvable when the input graph is a tournament (i.e., a directed complete graph) (Chakrabarti et al. [SODA 2020]; Baweja, Jia, and Woddruff [ITCS 2022]), or close to a tournament in edit distance (Ghosh and Kuchlous [ESA 2024]). Extending this line of work, our main result shows, at a qualitative level, that the streaming complexity of strong connectivity certificates and related problems is parameterized by independence number, demonstrating a continuum of hardness for directed graph connectivity problems. Quantitatively, for an $n$-node graph with independence number $α$, we give $p$-pass randomized algorithms that compute a $k$-node strong connectivity certificate of size $O(αn)$ using $\tilde{O}(k^{1-1/p}αn^{1+1/p})$ space in the insertion-only model. For the lower bound, we show that even when $k=1$, any $p$-pass streaming algorithm for a 1-node strong connectivity certificate in the insertion-only model requires $Ω(αn/p)$ space. To derive these lower bounds, we introduce the gadget-embedding tournament framework to construct direct-sum-type hard instances with a prescribed independence number, which is applicable to lower-bounding a wide range of directed graph problems.

翻译：本研究探讨在流式与分布式模型中，计算有向图稀疏子图以验证连通性的空间复杂度。形式化地，对于有向图 $G=(V,A)$ 及 $k\in \mathbb{N}$，一个 $k$ 节点强连通性证书是指满足以下条件的子图 $H=(V,A')\subseteq G$：对于任意两个不同节点 $s,t\in V$，$H$ 中从 $s$ 到 $t$ 的成对内部节点不相交路径数量至少为 $k$，或与 $G$ 中对应数量相同。鉴于有向连通性问题固有的困难性，先前多项研究聚焦于受限图类，表明当输入图为竞赛图（即有向完全图）（Chakrabarti 等人 [SODA 2020]；Baweja、Jia 和 Woddruff [ITCS 2022]）或在编辑距离上接近竞赛图时（Ghosh 和 Kuchlous [ESA 2024]），若干一般情况下的难题可被高效求解。沿此研究方向，我们的主要结果在定性层面表明，强连通性证书及相关问题的流式计算复杂度可由独立数参数化，从而揭示有向图连通性问题存在连续的困难度谱系。在定量层面，对于具有独立数 $α$ 的 $n$ 节点图，我们提出了 $p$ 轮随机算法，在仅插入模型中能以 $\tilde{O}(k^{1-1/p}αn^{1+1/p})$ 空间计算规模为 $O(αn)$ 的 $k$ 节点强连通性证书。在下界证明方面，我们表明即使当 $k=1$ 时，任何在仅插入模型中计算 1 节点强连通性证书的 $p$ 轮流式算法均需要 $Ω(αn/p)$ 空间。为推导这些下界，我们引入了构件嵌入竞赛图框架，用以构造具有指定独立数的直和型困难实例，该框架可广泛应用于各类有向图问题的下界证明。