On the Power of Multiplexing in Number-on-the-Forehead Protocols

We study the direct-sum problem for $k$-party ``Number On the Forehead'' (NOF) deterministic communication complexity. We prove several positive results, showing that the complexity of computing a function $f$ in this model, on $\ell$ instances, may be significantly cheaper than $\ell$ times the complexity of computing $f$ on a single instance. Quite surprisingly, we show that this is the case for ``most'' (boolean, $k$-argument) functions. We then formalize two-types of sufficient conditions on a NOF protocol $Q$, for a single instance, each of which guarantees some communication complexity savings when appropriately extending $Q$ to work on $\ell$ instances. One such condition refers to what each party needs to know about inputs of the other parties, and the other condition, additionally, refers to the communication pattern that the single-instance protocol $Q$ uses. In both cases, the tool that we use is ``multiplexing'': we combine messages sent in parallel executions of protocols for a single instance, into a single message for the multi-instance (direct-sum) case, by xoring them with each other.