Necessary and sufficient condition for graph-aware BFT consensus in granular asynchrony

Determine the necessary and sufficient condition on an undirected communication graph G=(V,E) under the granular asynchrony timing model—where links may be synchronous, partially synchronous, or asynchronous—that permits deterministic Byzantine fault-tolerant consensus by algorithms that know the underlying graph (i.e., are graph-aware), as opposed to graph-agnostic algorithms for which a sufficient condition is provided in the paper.

Background

The paper introduces granular synchrony, modeling networks as graphs with a mixture of synchronous, partially synchronous, and asynchronous links. It derives necessary and sufficient conditions for crash- and Byzantine-fault-tolerant consensus under variants of this model.

For the granular asynchrony variant in the Byzantine setting, the authors provide a requirement ensuring liveness for graph-agnostic algorithms: existence of a correct node with partially synchronous paths to at least f other correct nodes. They explicitly note that while such conditions are established for graph-agnostic algorithms, determining the tight necessary and sufficient condition for algorithms tailored to (and aware of) the specific communication graph remains unresolved.

This open question highlights a gap between algorithms that do not assume knowledge of the graph and those that can leverage such knowledge; resolving it would complete the characterization of BFT consensus feasibility under granular asynchrony.