For a fixed graph property $\Phi$ and integer $k \geq 1$, the problem $\#\mathrm{IndSub}(\Phi,k)$ asks to count the induced $k$-vertex subgraphs satisfying $\Phi$ in an input graph $G$. If $\Phi$ is trivial on $k$-vertex graphs (i.e., if $\Phi$ contains either all or no $k$-vertex graphs), this problem is trivial. Otherwise we prove, among other results: - If $\Phi$ is edge-monotone (i.e., closed under deleting edges), then $\#\mathrm{IndSub}(\Phi,k)$ cannot be solved in time $n^{o(k)}$ assuming ETH. This strengthens a result by D\"oring, Marx and Wellnitz [STOC 2024] that only ruled out an exponent of $o(\sqrt{\log k}/ \log \log k)$. Our results also hold when counting modulo fixed primes. - If there is some fixed $\varepsilon > 0$ such that at most $(2-\varepsilon)^{\binom{k}{2}}$ graphs on $k$ vertices satisfy $\Phi$, then $\#\mathrm{IndSub}(\Phi,k)$ cannot be solved in time $n^{o(k/\sqrt{\log k})}$ assuming ETH. Our results hold even when each of the graphs in $\Phi$ may come with an arbitrary individual weight. This generalizes previous results for hereditary properties by Focke and Roth [SIAM J.\ Comput.\ 2024] up to a $\sqrt{\log k}$ factor in the exponent of the lower bound. - If $\Phi$ only depends on the number of edges, then $\#\mathrm{IndSub}(\Phi,k)$ cannot be solved in time $n^{o(k)}$ assuming ETH. This improves on a lower bound by Roth, Schmitt and Wellnitz [FOCS 2020] that only ruled out an exponent of $o(k / \sqrt{\log k})$. In all cases, we also obtain $\mathsf{\#W[1]}$-hardness if $k$ is part of the input and the problem is parameterized by $k$. We also obtain lower bounds on the Weisfeiler-Leman dimension. Our results follow from relatively straightforward Fourier analysis, and our paper subsumes most of the known $\mathsf{\#W[1]}$-hardness results known in the area, often with tighter lower bounds under ETH.

翻译：对于一个固定的图性质 $\Phi$ 和整数 $k \geq 1$，问题 $\#\mathrm{IndSub}(\Phi,k)$ 要求计算输入图 $G$ 中满足 $\Phi$ 的 $k$ 顶点诱导子图的数量。如果 $\Phi$ 在 $k$ 顶点图上是平凡的（即 $\Phi$ 要么包含所有 $k$ 顶点图，要么不包含任何 $k$ 顶点图），那么该问题是平凡的。否则，我们证明了以下结果（包括其他结果）：- 如果 $\Phi$ 是边单调的（即在删除边操作下封闭），则在指数时间假设（ETH）下，$\#\mathrm{IndSub}(\Phi,k)$ 无法在 $n^{o(k)}$ 时间内求解。这强化了 D\"oring、Marx 和 Wellnitz [STOC 2024] 的结果，后者仅排除了 $o(\sqrt{\log k}/ \log \log k)$ 的指数。我们的结果在模固定素数计数时也成立。- 如果存在某个固定的 $\varepsilon > 0$，使得最多有 $(2-\varepsilon)^{\binom{k}{2}}$ 个 $k$ 顶点图满足 $\Phi$，则在 ETH 假设下，$\#\mathrm{IndSub}(\Phi,k)$ 无法在 $n^{o(k/\sqrt{\log k})}$ 时间内求解。我们的结果甚至在 $\Phi$ 中的每个图可能带有任意独立权重时也成立。这将 Focke 和 Roth [SIAM J.\ Comput.\ 2024] 关于遗传性质的结果推广至下界指数中的 $\sqrt{\log k}$ 因子范围内。- 如果 $\Phi$ 仅依赖于边的数量，则在 ETH 假设下，$\#\mathrm{IndSub}(\Phi,k)$ 无法在 $n^{o(k)}$ 时间内求解。这改进了 Roth、Schmitt 和 Wellnitz [FOCS 2020] 的下界，后者仅排除了 $o(k / \sqrt{\log k})$ 的指数。在所有情况下，如果 $k$ 是输入的一部分且问题以 $k$ 为参数，我们还得到了 $\mathsf{\#W[1]}$-困难性。我们还得到了关于 Weisfeiler-Leman 维度的下界。我们的结果源于相对直接的傅里叶分析，并且本文涵盖了该领域大多数已知的 $\mathsf{\#W[1]}$-困难性结果，通常在 ETH 下具有更紧的下界。