Problem 72: Communication Complexity of Approximating Set-Intersection Join

Suggested by	Qin Zhang
Source	Baltimore 2016
Short link	https://sublinear.info/72

For $\varepsilon\in(0,1]$ and $n\geq 1$, consider the following communication complexity problem $\mathrm{SIJ}_{n,\varepsilon}$: Alice and Bob are respectively given matrices $A,B\in\{0,1\}^{n\times n}$, and wish to approximate the number of non-zero elements of the product $C=AB$: i.e., to output an $(1+\varepsilon)$-approximation of $\operatorname{nnz}(C)$. What is the two-way randomized communication complexity $R_\delta(\mathrm{SIJ}_{n,\varepsilon})$ (for probability of error $\delta$?)

What is known [GuchtWWZ-15]:

- $R^{\to}_{1/n}(\mathrm{SIJ}_{n,\varepsilon}) = \tilde{O}(\frac{n}{\varepsilon^2})$ (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.