We study the $k$-edge connectivity problem on undirected graphs in the distributed sketching model, where we have $n$ nodes and a referee. Each node sends a single message to the referee based on its 1-hop neighborhood in the graph, and the referee must decide whether the graph is $k$-edge connected by taking into account the received messages. We present the first lower bound for deciding a graph connectivity problem in this model with a deterministic algorithm. Concretely, we show that the worst case message length is $Ω( k )$ bits for $k$-edge connectivity, for any super-constant $k = O(\sqrt{n})$. Previously, only a lower bound of $Ω( \log^3 n )$ bits was known for ($1$-edge) connectivity, due to Yu (SODA 2021). In fact, our result is the first super-polylogarithmic lower bound for a connectivity decision problem in the distributed graph sketching model. To obtain our result, we introduce a new lower bound graph construction, as well as a new 3-party communication complexity problem that we call UniqueOverlap. As this problem does not appear to be amenable to reductions to existing hard problems such as set disjointness or indexing due to correlations between the inputs of the three players, we leverage results from cross-intersecting set families to prove the hardness of UniqueOverlap for deterministic algorithms. Finally, we obtain the sought lower bound for deciding $k$-edge connectivity via a novel simulation argument that, in contrast to previous works, does not introduce any probability of error and thus works for deterministic algorithms.

翻译：本文研究无向图上$k$-边连通性问题的分布式草图模型，该模型包含$n$个节点和一个裁判节点。每个节点根据其在图中的1跳邻域信息向裁判节点发送单条消息，裁判节点必须依据接收到的消息判定该图是否具有$k$-边连通性。我们首次在该模型中针对确定性算法给出了图连通性问题的下界。具体而言，我们证明对于任意超常数$k = O(\sqrt{n})$，判定$k$-边连通性所需的最坏情况消息长度为$Ω( k )$比特。此前，由于Yu的研究（SODA 2021），仅知（1-边）连通性的下界为$Ω( \log^3 n )$比特。事实上，我们的结果是分布式图草图模型中连通性判定问题的首个超多对数下界。为获得该结果，我们引入了新的下界图构造方法，以及一个称为UniqueOverlap的新型三方通信复杂度问题。由于该问题中三方参与者的输入存在相关性，难以归约至集合互斥或索引检索等现有难题，我们利用交叉相交集族的研究结果证明了UniqueOverlap对确定性算法的困难性。最终，通过新颖的模拟论证方法，我们获得了判定$k$-边连通性所需的下界。与先前研究不同，该论证方法不引入任何错误概率，因此适用于确定性算法。