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 R(f). The certificate complexity C(f) is such a complexity measure for the zero-error randomized query complexity R0(f): C(f) ≤R0(f) ≤C(f)2. In the first part of the paper we introduce a new complexity measure, expectational certificate complexity EC(f), which is also a quadratically tight bound on R0(f): EC(f) ≤R0(f) = O(EC(f)2). For R(f), we prove that EC2/3 ≤R(f). We then prove that EC(f) ≤C(f) ≤EC(f)2 and show that there is a quadratic separation between the two, thus EC(f) gives a tighter upper bound for R0(f). The measure is also related to the fractional…