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 C1(f) ≤ bs0(f)/2s0(f), achieved by Kenyon and Kutin [6]. We improve this to C1(f) ≥ 3bs0(f)/2s0(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. © 2014 Springer-Verlag Berlin Heidelberg.