Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

We investigate the problem $\#\mathrm{IndSub}\left(\Phi \right)$ of counting all induced subgraphs of size k in a graph G that satisfy a given property $\Phi$ . This continues the work of Jerrum and Meeks who proved the problem to be $\#W\left[1\right]$ -hard for some families of properties which include (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties $\Phi$ , the problem $\#\mathrm{IndSub}\left(\Phi \right)$ is hard for $\#W\left[1\right]$ if the reduced Euler characteristic of the associated simplicial (graph) complex of $\Phi$ is non-zero. This observation links $\#\mathrm{IndSub}\left(\Phi \right)$ to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that $\#\mathrm{IndSub}\left(\Phi \right)$ is $\#W\left[1\right]$ -hard for every monotone property $\Phi$ that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for $k>2$ . Moreover, we show that for those properties $\#\mathrm{IndSub}\left(\Phi \right)$ can not be solved in time $f\left(k\right)·n^{o\left(k\right)}$ for any computable function f unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that $\#\mathrm{IndSub}\left(\Phi \right)$ is $\#W\left[1\right]$ -hard if $\Phi$ is any non-trivial modularity constraint on the number of edges with respect to some prime q or if $\Phi$ enforces the presence of a fixed isolated subgraph.