Volume 9 (2013) Article 20 pp. 653-663
Approximating the AND-OR Tree
The approximate degree of a Boolean function $f$ is the least degree of a real polynomial that approximates $f$ within $1/3$ at every point. We prove that the function $\bigwedge_{i=1}^n\bigvee_{j=1}^nx_{ij}$, known as the AND-OR tree, has approximate degree $\Omega(n)$. This lower bound is tight and closes a line of research on the problem, the best previous bound being $\Omega(n^{0.75})$. More generally, we prove that the function $\bigwedge_{i=1}^m\bigvee_{j=1}^nx_{ij}$ has approximate degree $\Omega(\sqrt{mn}),$ which is tight. The same lower bound was obtained independently by Bun and Thaler (2013) using related techniques.