Quadratically Tight Relations for Randomized Query Complexity

In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions \(\mathsf {R}(f)\). The certificate complexity \(\mathsf {C}(f)\) is such a complexity measure for the zero-error randomized query complexity \(\mathsf {R}_0(f)\): \(\mathsf {C}(f) \le \mathsf {R}_0(f) \le \mathsf {C}(f)^2\). In the first part of the paper we introduce a new complexity measure, expectational certificate complexity \(\mathsf {EC}(f)\), which is also a quadratically tight bound on \(\mathsf {R}_0(f)\): \(\mathsf {EC}(f) \le \mathsf {R}_0(f) = O(\mathsf {EC}(f)^2)\). For \(\mathsf {R}(f)\), we prove that \(\mathsf {EC}^{2/3} \le \mathsf {R}(f)\). We then prove that \(\mathsf {EC}(f) \le \mathsf {C}(f) \le \mathsf {EC}(f)^2\) and show that there is a quadratic separation between the two, thus \(\mathsf {EC}(f)\) gives a tighter upper bound for \(\mathsf {R}_0(f)\). The measure is also related to the fractional certificate complexity \(\mathsf {FC}(f)\) as follows: \(\mathsf {FC}(f) \le \mathsf {EC}(f) = O(\mathsf {FC}(f)^{3/2})\). This also connects to an open question by Aaronson whether \(\mathsf {FC}(f)\) is a quadratically tight bound for \(\mathsf {R}_0(f)\), as \(\mathsf {EC}(f)\) is in fact a relaxation of \(\mathsf {FC}(f)\).