Chen and Flum showed that any FPT-approximation of the
k-
Clique problem is not in para-
and the
k-
DominatingSet (
k-
DomSet) problem could not be computed by para-
circuits. It is natural to
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Chen and Flum showed that any FPT-approximation of the
k-
Clique problem is not in para-
and the
k-
DominatingSet (
k-
DomSet) problem could not be computed by para-
circuits. It is natural to ask whether the
-approximation of the
k-
DomSet problem is in para-
for some computable function
f. Very recently it was proved that assuming
, the
k-
DomSet problem cannot be
-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-
circuits could not approximate this problem with ratio
for any computable function
f. Moreover, under the hypothesis that the 3-CNF-SAT problem cannot be computed by constant-depth circuits of size
for some
, we show that constant-depth circuits of size
cannot distinguish graphs whose dominating numbers are either ≤
k or >
. However, we find that the hypothesis may be hard to settle by showing that it implies
.
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