Counting Small Induced Subgraphs with Hereditary Properties

We study the computational complexity of the problem $\#\text{IndSub}(\Phi)$ of counting $k$-vertex induced subgraphs of a graph $G$ that satisfy a graph property $\Phi$. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH): - If a hereditary property $\Phi$ is true for all graphs, or if it is true only for finitely many graphs, then $\#\text{IndSub}(\Phi)$ is solvable in polynomial time. - Otherwise, $\#\text{IndSub}(\Phi)$ is $\#\mathsf{W[1]}$-complete when parameterised by $k$, and, assuming ETH, it cannot be solved in time $f(k)\cdot |G|^{o(k)}$ for any function $f$. This classification features a wide range of properties for which the corresponding detection problem (as classified by Khot and Raman [TCS 02]) is tractable but counting is hard. Moreover, even for properties which are already intractable in their decision version, our results yield significantly stronger lower bounds for the counting problem. As additional result, we also present an exhaustive and explicit parameterised complexity classification for all properties that are invariant under homomorphic equivalence. By covering one of the most natural and general notions of closure, namely, closure under vertex-deletion (hereditary), we generalise some of the earlier results on this problem. For instance, our results fully subsume and strengthen the existing classification of $\#\text{IndSub}(\Phi)$ for monotone (subgraph-closed) properties due to Roth, Schmitt, and Wellnitz [FOCS 20].