Approximate Agreement ($\mathcal{AA}$) is a fundamental primitive that, even in the presence of Byzantine faults, allows honest parties to obtain close (but not necessarily identical) outputs that lie within the range of their inputs. While the optimal round complexity of synchronous $\mathcal{AA}$ on real values is well understood, its extension to other input spaces has remained open, with fundamental questions regarding achievable resilience and round efficiency still unresolved. In this work, we investigate the optimal round complexity of synchronous $\mathcal{AA}$ on trees under Byzantine failures. In this setting, parties hold as inputs vertices of a publicly known labeled tree $T$ and must output $1$-close vertices lying in the convex hull of the honest inputs. We present a synchronous protocol with optimal resilience and round complexity $O\left(\frac{\log D(T)}{\log \log D(T)}\right)$, where $D(T)$ denotes the diameter of the input space tree. Complementing this result, we extend impossibility results for real-valued $\mathcal{AA}$ to any graph $G$ by proving a lower bound of $Ω\left(\frac{\log D(G)}{\log \log D(G) + \log \frac{n+t}{t}}\right)$ rounds, where $n$ is the number of parties and $t$ the number of Byzantine faults. Together, these results establish the asymptotic optimality of our protocol whenever $t \in Θ(n)$. We further extend our techniques to block graphs by leveraging their clique tree structure. This yields protocols for $\mathcal{AA}$ on block graphs with optimal resilience in both the synchronous and asynchronous models, and with optimal round complexity in the synchronous model.

翻译：近似一致（$\mathcal{AA}$）是一项基础原语，即使存在拜占庭故障，也能让诚实的参与方获得接近于（但不一定完全相同）其输入范围的输出。虽然实数值同步$\mathcal{AA}$的最优轮复杂度已被充分理解，但其在其他输入空间上的扩展仍悬而未决，关于可达弹性与轮效率的基本问题尚未解决。本文研究树结构下同步$\mathcal{AA}$在拜占庭故障场景中的最优轮复杂度。在该设定中，参与方持有公开标记树$T$的顶点作为输入，必须输出位于诚实输入凸包内且距离为$1$的顶点。我们提出同步协议，在最优弹性下达到轮复杂度$O\left(\frac{\log D(T)}{\log \log D(T)}\right)$，其中$D(T)$表示输入空间树的直径。作为补充，我们通过证明下界$\Omega\left(\frac{\log D(G)}{\log \log D(G) + \log \frac{n+t}{t}}\right)$轮（这里$n$为参与方数量，$t$为拜占庭故障数）将实数值$\mathcal{AA}$的不可能性结果扩展至任意图$G$。这些结果共同确立了在$t \in \Theta(n)$条件下协议渐近最优性。进一步，我们利用块图的团树结构扩展技术，为块图上的$\mathcal{AA}$分别设计同步和异步模型中具有最优弹性、同步模型中具有最优轮复杂度的协议。