# Problem 72: Communication Complexity of Approximating Set-Intersection Join

For $\varepsilon\in(0,1]$ and $n\geq 1$, consider the following communication complexity problem $\mathrm{SIJ}_{n,\varepsilon}$: Alice and Bob are given matrices $A,B\in\{0,1\}^{n\times n}$, respectively, and wish to output a $(1+\varepsilon)$-approximation to the number of non-zero entries in the product $C=AB$. What is the two-way randomized communication complexity $R_\delta(\mathrm{SIJ}_{n,\varepsilon})$ (where $\delta$ is the probability of error)?
• $R^{\to}_{1/n}(\mathrm{SIJ}_{n,\varepsilon}) = \tilde{O}(\frac{n}{\varepsilon^2})$ (one-way communication),
• $R_\delta(\mathrm{SIJ}_{n,\varepsilon}) = \Omega(\frac{n}{\varepsilon^{2/3}})$ for some absolute constant $\delta > 0$.
What is the right dependence on $\varepsilon$?
Note: $\mathrm{SIJ}$ stands for “Set-Intersection Join,” which is the motivation for this question.