AbstractWe study the problem $$\#\textsc {IndSub}(\varPhi )$$
#
I
N
D
S
U
B
(
Φ
)
of counting all induced subgraphs of size k in a graph G that satisfy the property $$\varPhi $$
Φ
. It is shown that, given any graph property $$\varPhi $$
Φ
that distinguishes independent sets from bicliques, $$\#\textsc {IndSub}(\varPhi )$$
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N
D
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B
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is hard for the class $$\#\mathsf {W[1]}$$
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W
[
1
]
, i.e., the parameterized counting equivalent of $${{\mathsf {N}}}{{\mathsf {P}}}$$
N
P
. Under additional suitable density conditions on $$\varPhi $$
Φ
, satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen $$\#\mathsf {W[1]}$$
#
W
[
1
]
-hardness by establishing that $$\#\textsc {IndSub}(\varPhi )$$
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N
D
S
U
B
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Φ
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cannot be solved in time $$f(k)\cdot n^{o(k)}$$
f
(
k
)
·
n
o
(
k
)
for any computable function f, unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime.