Problem 72: Communication Complexity of Approximating Set-Intersection Join

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Suggested by Qin Zhang
Source Baltimore 2016
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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)?

Known facts [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.