The Inapproximability of k-DominatingSet for Parameterized $\mathsf {{AC}^0}$ Circuits

Abstract

In [4], Chen and Flum showed that any FPT-approximation of the $k$ -Clique problem is not in ${para- \mathsf {AC}}^{0}$ and the $k$ -DominatingSet ($k$-DomSet) problem could not be computed by ${para- \mathsf {AC}}^{0}$ circuits. It is natural to ask whether f(k)-FPT-approximation of the $k$-DomSet problem is in ${para- \mathsf {AC}}^{0}$ for some computable function f.

Very recently [13, 20] showed that assuming $\mathsf {W[1]}\ne \mathsf {FPT}{}$, the $k$-DomSet cannot be approximated by FPT algorithms. We observe that the constructions in [13] can be carried out in ${para- \mathsf {AC}}^{0}$, and thus we prove that ${para- \mathsf {AC}}^{0}$ circuits could not approximate this problem with ratio f(k) for any computable function f. Moreover, under the hypothesis that the 3-CNF-SAT problem cannot be computed by constant-depth circuits of size $2^{\varepsilon n}$ for some $\varepsilon >0$, we show that constant-depth circuits of size $n^{o(k)}$ cannot distinguish graphs whose dominating numbers are either $\le k$ or $>\root k \of { \frac{\log n}{3\log \log n} }$. However, we find that the hypothesis may be hard to settle by showing that it implies $\mathsf {NP}\not \subseteq \mathsf {NC^1}$.