Parity Decision Tree Complexity and 4-Party Communication Complexity of XOR-functions Are Polynomially Equivalent

In this note, we study the relation between the parity decision tree complexity of a boolean function $f$, denoted by $\mathrm{D}_{\oplus}(f)$, and the $k$-party number-in-hand multiparty communication complexity of the XOR functions $F(x_1,\ldots, x_k)= f(x_1\oplus\cdots\oplus x_k)$, denoted by $\mathrm{CC}^{(k)}(F)$. It is known that $\mathrm{CC}^{(k)}(F)\leq k\cdot\mathrm{D}_{\oplus}(f)$ because the players can simulate the parity decision tree that computes $f$. In this note, we show that $\mathrm{D}_{\oplus}(f)\leq O\big(\mathrm{CC}^{(4)}(F)^5\big).$ Our main tool is a recent result from additive combinatorics due to Sanders. As $\mathrm{CC}^{(k)}(F)$ is non-decreasing as $k$ grows, the parity decision tree complexity of $f$ and the communication complexity of the corresponding $k$-argument XOR functions are polynomially equivalent whenever $k\geq 4$. Remark: After the first version of this paper was finished, we discovered that Hatami and Lovett had already discovered the same result a few years ago, without writing it up.

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