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This paper demonstrates that the problem of communication complexity is NP-hard, implying significant challenges in computational complexity.
In the paper where he first defined Communication Complexity, Yao asks: Is computing $CC(f)$ (the 2-way communication complexity of a given function f) NP-complete? The problem of deciding whether CC(f) ≤ k, when given the communication matrix for f and a number k, is easily seen to be in NP. Kushilevitz and Weinreb have shown that this problem is cryptographically hard. Here we show it is NP-hard.
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Hirahara et al. (2025) studied this question.
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