Round and Resilience-Optimal Approximate Agreement on Trees and Block Graphs

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 $\Omega\left(\frac{\log D(G)}{\log \log D(G) + \log \frac{n+t}{t}}\right)$ 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 $t \in \Theta(n)$. 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.