Towards Better Separation between Deterministic and Randomized Query Complexity

TL;DR

This paper demonstrates new separations between deterministic and randomized query complexities for a specific Boolean function, refuting a previous conjecture and highlighting the widest known gap between certain complexity measures.

Contribution

It establishes the first separation results for the original function $F$, showing that $R_1(F)$ can be significantly smaller than $D(F)$, challenging prior conjectures.

Findings

$R_1(F) = ilde{O}( oot 2 ext{ of } D(F))$

$R_0(F) = ilde{O}(D(F)^{3/4})$

Refutes Saks and Wigderson's conjecture on the lower bound of $R_0(f)$

Abstract

We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$ and randomized one-sided error query complexity $(R_1(F))$: $R_1(F) = \widetilde{O}(\sqrt{D(F)})$ and $R_0(F)=\widetilde{O}(D(F))^{3/4}$. This refutes the conjecture made by Saks and Wigderson that for any Boolean function $f$, $R_0(f)=\Omega({D(f)})^{0.753..}$. This also shows widest separation between $R_1(f)$ and $D(f)$ for any Boolean function. The function $F$ was defined by G{\"{o}}{\"{o}}s, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function $F$ certify optimal (quadratic) separation between $D(f)$ and $R_0(f)$, and polynomial separation between $R_0(f)$ and $R_1(f)$. Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considerd in the work of Ambainis et al are different variants of $F$, we work with the original function $F$ itself.