Computer Science/Discrete Mathematics Seminar I

I will discuss a lower bound of $\omega(n^(k/4))$ on the size of constant-depth $(AC_0)$ circuits solving the k-clique problem on n-vertex graphs. This bound follows from a stronger result that $AC_0$ circuits of size $O(n^(k/4))$ almost surely fail to distinguish between an Erdos-Renyi random graph $G$ at the k-clique threshold and $G' = G \cup$(random k-clique). This lower bound is nearly matched by a recent construction of Amano of $AC_0$ circuits of size $O(n^((k/4)+const.))$ which almost surely distinguish $G$ from $G'$ . I will also discuss a corollary in logic: strictness of the "variable hierarchy" for first-order logic on ordered graphs.