Independence-Number Parameterized Space Complexity for Directed Connectivity Certificate

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 $\alpha$, we give $p$-pass randomized algorithms that compute a $k$-node strong connectivity certificate of size $O(\alpha n)$ using $\tilde{O}(k^{1-1/p}\alpha n^{1+1/p})$ 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 $\Omega(\alpha n/p)$ 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.