A little advice can be very helpful
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R∈(f) and $$D^\mu_\in (f)$$denote the randomized and μ-distributional communication complexities of f, respectively (∈ a small constant). Yao’s well-known minimax principle states that $$R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}$$. Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function $$f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\}$$for which maxμ product $$\{D^\mu_\in (f)\} = \Theta(1) \,{\textrm but}\, R_{\in} (f) = \Theta(n)$$. We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).