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### Paper:

TR21-030 | 2nd March 2021 19:54

#### Hardness of Constant-round Communication Complexity

TR21-030
Authors: Shuichi Hirahara, Rahul Ilango, Bruno Loff
Publication: 2nd March 2021 23:31
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.