The direct sum of universal relations

The universal relation is the communication problem in which Alice and Bob get as inputs two distinct strings, and they are required to find a coordinate on which the strings differ. The study of this problem is motivated by its connection to Karchmer–Wigderson relations [12], which are communication problems that are tightly related to circuit-depth lower bounds. In this paper, we prove a direct sum theorem for the universal relation, namely, we prove that solving m independent instances of the universal relation is m times harder than solving a single instance. More specifically, it is known that the deterministic communication complexity of the universal relation is at least n. We prove that the deterministic communication complexity of solving m independent instances of the universal relation is at least m⋅(n−O(log⁡m)).