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Medientyp:
E-Artikel
Titel:
The Inapproximability of k-DominatingSet for Parameterized
AC
0
Circuits †
Beteiligte:
Lai, Wenxing
Erschienen:
MDPI AG, 2019
Erschienen in:
Algorithms, 12 (2019) 11, Seite 230
Sprache:
Englisch
DOI:
10.3390/a12110230
ISSN:
1999-4893
Entstehung:
Anmerkungen:
Beschreibung:
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 .