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

  • Wenxing LaiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11458)

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 \(\mathsf {NP}\not \subseteq \mathsf {NC^1}\).

Keywords

Parameterized \(\mathsf {{AC}^0}\) Dominating set Inapproximability 

Notes

Acknowledgement

I am grateful to Yijia Chen, Bundit Laekhanukit and Chao Liao for many helpful discussions and valuable comments. I also thank the anonymous referees for their detailed comments. This research is supported by National Natural Science Foundation of China (Project 61872092).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Intelligent Information Processing, School of Computer ScienceFudan UniversityShanghaiChina

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