Augustine et al. [DISC 2022] initiated the study of distributed graph algorithms in the presence of Byzantine nodes in the congested clique model. In this model, there is a set $B$ of Byzantine nodes, where $|B|$ is less than a third of the total number of nodes. These nodes have complete knowledge of the network and the state of other nodes, and they conspire to alter the output of the system. The authors addressed the connectivity problem, showing that it is solvable under the promise that either the subgraph induced by the honest nodes is connected, or the graph has $2|B|+1$ connected components. In the current work, we continue the study of the Byzantine congested clique model by considering the recognition of other graph properties, specifically \emph{hereditary properties}. A graph property is \emph{hereditary} if it is closed under taking induced subgraphs. Examples of hereditary properties include acyclicity, bipartiteness, planarity, and bounded (chromatic, independence) number, etc. For each class of graphs $\mathcal{G}$ satisfying an hereditary property (an hereditary graph-class), we propose a randomized algorithm which, with high probability, (1) accepts if the input graph $G$ belongs to $\mathcal{G}$, and (2) rejects if $G$ contains at least $|B| + 1$ disjoint subgraphs not belonging to $\mathcal{G}$. The round complexity of our algorithm is $$\mathcal{O}\left(\left(\dfrac{\log \left(\left|\mathcal{G}_n\right|\right)}{n} +|B|\right)\cdot\textrm{polylog}(n)\right),$$ where $\mathcal{G}_n$ is the set of $n$-node graphs in $\mathcal{G}$. Finally, we obtain an impossibility result that proves that our result is tight. Indeed, we consider the hereditary class of acyclic graphs, and we prove that there is no algorithm that can distinguish between a graph being acyclic and a graph having $|B|$ disjoint cycles.

翻译：Augustine等人[DISC 2022]在拥塞团模型下开创性地研究了拜占庭节点存在时的分布式图算法。在该模型中，存在一个拜占庭节点集合$B$，且$|B|$小于总节点数的三分之一。这些节点完全了解网络及其他节点的状态，并合谋篡改系统输出。作者解决了连通性问题，证明在满足以下承诺时可求解：要么诚实节点诱导的子图是连通的，要么该图具有$2|B|+1$个连通分量。本文继续研究拜占庭拥塞团模型，考虑对其他图性质的识别，特别是\emph{遗传性质}。一个图性质是\emph{遗传的}，如果它在取诱导子图下封闭。遗传性质的示例包括无环性、二部性、平面性以及有界（色数、独立数）等。对于每个满足遗传性质的图类$\mathcal{G}$（遗传图类），我们提出一种随机化算法，该算法能以高概率：（1）若输入图$G$属于$\mathcal{G}$则接受；（2）若$G$包含至少$|B|+1$个不属于$\mathcal{G}$的不相交子图则拒绝。算法的轮复杂度为$$\mathcal{O}\left(\left(\dfrac{\log \left(\left|\mathcal{G}_n\right|\right)}{n} +|B|\right)\cdot\textrm{polylog}(n)\right),$$其中$\mathcal{G}_n$是$\mathcal{G}$中$n$节点图的集合。最后，我们得到不可行性结果，证明我们的结果是紧的。具体而言，我们考虑无环图的遗传类，并证明不存在能够区分图是无环图与图具有$|B|$个不相交环路的算法。