Some lower bounds in parameterized ${\mathrm{AC}}^0$

Abstract

We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical ${\mathrm{AC}}^0$. Among others, we derive such a lower bound for all fpt-approximations of the parameterized clique problem and for a parameterized halting problem, which recently turned out to link problems of computational complexity, descriptive complexity, and proof theory. To show the first lower bound, we prove a strong ${\mathrm{AC}}^0$ version of the planted clique conjecture: ${\mathrm{AC}}^0$-circuits asymptotically almost surely can not distinguish between a random graph and this graph with a randomly planted clique of any size $\le n^{\xi }$ (where $0\le \xi <1$).