Worst Case Nonzero-Error Interactive Communication

In the interactive communication model, two parties $P_{\cal X}$ and $P_{\cal Y}$ possess respective private but correlated inputs $x$ and $y$, and $P_{\cal Y}$ wants to learn $x$ from $P_{\cal X}$ while minimizing the communication required for the worst possible input pair $(x,y)$. Our contribution is the analysis of four nonzero-error models in this correlated data setting. In the private coin randomized model, both players are allowed to toss coins, and $P_Y$ must learn $x$ with high probability for every input pair. The second and third models are similar to the first one, but the players are allowed to use a common source of randomness and to solve several independent instances of the same problem simultaneously, respectively. In the fourth model, $P_{\cal Y}$ is allowed to answer incorrectly for a small fraction of the inputs. We show that one round of communication is nearly optimal for the private coin randomized model. We also prove that the last three models are equivalent and can be arbitrarily better than the original worst case deterministic model when interaction is not allowed. Finally, we show that the deterministic model and all the nonzero-error models are equivalent for a class of symmetric problems arising from several practical applications, although nonzero-error and randomization allow efficient one-way protocols.