# 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.

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