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# The Inapproximability of k-DominatingSet for Parameterized AC 0 Circuits †

by Wenxing Lai Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 201203, China
This paper is an extended version of our paper published in the Thirteenth International Frontiers of Algorithmics Workshop (FAW2019), Sanya, China, 29 April–3 May 2019.
Algorithms 2019, 12(11), 230; https://doi.org/10.3390/a12110230
Received: 18 July 2019 / Revised: 20 September 2019 / Accepted: 11 October 2019 / Published: 4 November 2019
(This article belongs to the Special Issue New Frontiers in Parameterized Complexity and Algorithms)
Chen and Flum showed that any FPT-approximation of the k-Clique problem is not in para- $AC 0$ and the k-DominatingSet (k-DomSet) problem could not be computed by para- $AC 0$ circuits. It is natural to ask whether the $f ( k )$ -approximation of the k-DomSet problem is in para- $AC 0$ for some computable function f. Very recently it was proved that assuming $W [ 1 ] ≠ FPT$ , the k-DomSet problem cannot be $f ( k )$ -approximated by FPT algorithms for any computable function f by S., Laekhanukit and Manurangsi and Lin, seperately. We observe that the constructions used in Lin’s work can be carried out using constant-depth circuits, and thus we prove that para- $AC 0$ 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 $2 ε n$ for some $ε > 0$ , we show that constant-depth circuits of size $n o ( k )$ cannot distinguish graphs whose dominating numbers are either ≤k or > $log n 3 log log n 1 / k$ . However, we find that the hypothesis may be hard to settle by showing that it implies $NP ⊈ NC 1$ . View Full-Text
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Lai, W. The Inapproximability of k-DominatingSet for Parameterized AC 0 Circuits . Algorithms 2019, 12, 230.