Optimal consensus error probability for arbitrary parameters

Determine the optimal error probability of deterministic consensus algorithms in the stochastic broadcast model for every number of processes n, every number of rounds r, and every broadcast-success probability p in [0,1].

Background

The paper studies deterministic binary consensus in the stochastic broadcast model, where each process broadcasts synchronously and each broadcast independently succeeds with probability p. Because message delivery is stochastic and senders do not learn whether their own broadcasts succeeded, consensus cannot be solved with zero error; the objective is instead to minimize the worst-case probability of disagreement.

The authors establish exact one-round optimality results for three processes and for certain parameter ranges when n is arbitrary, as well as a communication-optimal multi-round construction. They do not determine the minimum error probability for arbitrary combinations of n, r, and p. They reduce the one-round optimization problem to minimum-weight cuts in reduced Kripke graphs and indicate that the corresponding multi-round optimization remains unresolved.