Hardness of constant-round communication complexity

How difficult is it to compute the communication complexity of a two-argument total Boolean function f : [N] X [N] → {0, 1}, when it is given as an N X N binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard. In this work, we show that it is NP-hard to approximate the size (number of leaves) of the smallest constant-round protocol for a two-argument total Boolean function f : [N] X [N] → {0, 1}, when it is given as an N X N binary matrix. Along the way to proving this, we show a new deterministic variant of the round elimination lemma, which may be of independent interest.