How difficult is it to compute the communication complexity of a two-argument total Boolean function $f:[N]\times[N]\to\{0,1\}$, when it is given as an $N\times N$ 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 $f:[N]\times[N]\to\{0,1\}$, when it is given as an $N\times N$ 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.