📝 SummarXiv

Abstract

Approximate Agreement (AA) is a key consensus 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 AA on real values is well understood, its extension to other input spaces remains an open problem. Our work is concerned with AA on trees, where the parties hold as inputs vertices from a publicly known labeled tree $T$ and must output $1$-close vertices in the honest inputs' convex hull. We present a protocol in the synchronous model, with optimal resilience and round complexity $O\left(\frac{\log D(T)}{\log\log D(T)}\right)$, where $D(T)$ is the diameter of the input space tree $T$. Our protocol relies on a simple reduction to real-valued AA. Additionally, we extend the impossibility results regarding the round complexity of synchronous $AA$ protocols on real values to trees: we prove a lower bound of $\Omega\left(\frac{\log D(T)}{\log \log D(T) + \log \frac{n + t}{t}} \right)$ rounds, where $n$ denotes the number of parties, and $t$ denotes the number of Byzantine parties. This establishes the asymptotic optimality of our protocol.