On Separation Between the Degree of a Boolean Function and the Block Sensitivity

Abstract

In this paper we study the separation between two complexity measures: the degree of a Boolean function as a polynomial over the reals and the block sensitivity. We show that the upper bound on the largest possible separation between these two measures can be improved from d2(f) ≥ bs(f), established by Tal [19], to d2(f)≥(10-2)bs(f). As a corollary, we show that the similar upper bounds between some other complexity measures are not tight as well, for instance, we can improve the recent sensitivity conjecture result by Huang [10] s4(f) ≥ bs(f) to s4(f)≥(10-2)bs(f). Our techniques are based on the paper by Nisan and Szegedy [14] and include more detailed analysis of a symmetrization polynomial. In our next result we show the same type of improvement for the separation between the approximate degree of a Boolean function and the block sensitivity: we show that deg1/32(f)≥6/101bs(f) and improve the previous result by Nisan and Szegedy [14] deg1/3(f)≥bs(f)/6. In addition, we construct an example showing that the gap between the constants in the lower bound and in the known upper bound is less than 0.2. In our last result we study the properties of a conjectured fully sensitive function on 10 variables of degree 4, existence of which would lead to improvement of the biggest known gap between these two measures. We prove that there is the only univariate polynomial that can be achieved by symmetrization of such a function by using the combination of interpolation and linear programming techniques.