A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity

Abstract

Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy [7], is whether sensitivity and block sensitivity are polynomially related. Motivated by the constructions of functions which achieve the largest known separations, we study the relation between 1-certificate complexity and 0-sensitivity and 0-block sensitivity.

Previously the best known lower bound was $C_1(f)\geq \frac{bs_0(f)}{2 s_0(f)}$, achieved by Kenyon and Kutin [6]. We improve this to $C_1(f)\geq \frac{3 bs_0(f)}{2 s_0(f)}$. While this improvement is only by a constant factor, this is quite important, as it precludes achieving a superquadratic separation between bs(f) and s(f) by iterating functions which reach this bound. In addition, this bound is tight, as it matches the construction of Ambainis and Sun [3] up to an additive constant.

This research has received funding from the EU Seventh Framework Programme (FP7/2007-2013) under projects QALGO (No. 600700) and RAQUEL (No. 323970) and ERC Advanced Grant MQC. Part of this work was done while Andris Ambainis was visiting Institute for Advanced Study, Princeton, supported by National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.