Quadratically Tight Relations for Randomized Query Complexity

TL;DR

This paper introduces the expectational certificate complexity, a new measure that tightly bounds zero-error randomized query complexity and relates to existing measures, providing improved bounds and insights into query complexity.

Contribution

The paper defines expectational certificate complexity, proving its quadratic tightness with randomized query complexity and establishing its relation to other complexity measures.

Findings

EC(f) tightly bounds R_0(f) as O(EC(f)^2)

EC(f) relates to C(f) and FC(f) with quadratic and fractional bounds

Upper bounds distributed query complexity using the square of the corruption bound

Abstract

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is also a quadratically tight bound on $R_0(f)$: $EC(f) \leq R_0(f) = O(EC(f)^2)$. We prove that $EC(f) \leq C(f) \leq EC(f)^2$ and show that there is a quadratic separation between the two, thus $EC(f)$ gives a tighter upper bound for $R_0(f)$. The measure is also related to the fractional certificate complexity $FC(f)$ as follows: $FC(f) \leq EC(f) = O(FC(f)^{3/2})$. This also connects to an open question by Aaronson whether $FC(f)$ is a quadratically tight bound for $R_0(f)$, as $EC(f)$ is in fact a relaxation of $FC(f)$. In the second part of the work, we upper bound the distributed query complexity $D^\mu_\epsilon(f)$ for product distributions $\mu$ by the square of the query corruption bound ($\mathrm{corr}_\epsilon(f)$) which improves upon a result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for communication complexity is open.