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

TL;DR

This paper establishes lower bounds for parameterized problems within the class ${ m AC}^0$, including the parameterized clique and halting problems, using circuit complexity and planted clique conjectures.

Contribution

It introduces new lower bounds for parameterized problems in ${ m AC}^0$, linking circuit complexity, parameterized complexity, and conjectures like planted clique.

Findings

${ m AC}^0$ circuits cannot distinguish certain random graphs with planted cliques.

Lower bounds are derived for all fpt-approximations of the parameterized clique problem.

A strong ${ m AC}^0$ version of the planted clique conjecture is proved.

Abstract

We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical ${\rm AC}^0$. 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 ${\rm AC}^0$ version of the planted clique conjecture: ${\rm AC}^0$-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 $0 \le \xi < 1$).