Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness
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We investigate the problem #IndSub(Φ) of counting all induced subgraphs of size k in a graph G that satisfy a given property Φ. This continues the work of Jerrum and Meeks who proved the problem to be #W[1]-hard for some families of properties which include, among others, (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 Φ, the problem #IndSub(Φ) is hard for #W[1] if the reduced Euler characteristic of the associated simplicial (graph) complex of Φ is non-zero. This observation links #IndSub(Φ) 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 #IndSub(Φ) is #W[1]-hard for every monotone property Φ 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 #IndSub(Φ) can not be solved in time f(k)· n^o(k) 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 #IndSub(Φ) is #W[1]-hard if Φ is any non-trivial modularity constraint on the number of edges with respect to some prime q or if Φ enforces the presence of a fixed isolated subgraph.
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