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

Abstract

AbstractWe investigate the problem $$\#{{\mathsf {IndSub}}}(\varPhi )$$ # IndSub ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy a given property $$\varPhi $$ Φ . This continues the work of Jerrum and Meeks who proved the problem to be $$\#{{\mathrm {W[1]}}}$$ # W [ 1 ] -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 $$\varPhi $$ Φ , the problem $$\#{{\mathsf {IndSub}}}(\varPhi )$$ # IndSub ( Φ ) is hard for $$\#{{\mathrm {W[1]}}}$$ # W [ 1 ] if the reduced Euler characteristic of the associated simplicial (graph) complex of $$\varPhi $$ Φ is non-zero. This observation links $$\#{{\mathsf {IndSub}}}(\varPhi )$$ # 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 $$\#{{\mathsf {IndSub}}}(\varPhi )$$ # IndSub ( Φ ) is $$\#{{\mathrm {W[1]}}}$$ # W [ 1 ] -hard for every monotone property $$\varPhi $$ 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 $$k > 2$$ k > 2 . Moreover, we show that for those properties $$\#{{\mathsf {IndSub}}}(\varPhi )$$ # IndSub ( Φ ) can not 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 (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that $$\#{{\mathsf {IndSub}}}(\varPhi )$$ # IndSub ( Φ ) is $$\#{{\mathrm {W[1]}}}$$ # W [ 1 ] -hard if $$\varPhi $$ Φ is any non-trivial modularity constraint on the number of edges with respect to some prime q or if $$\varPhi $$ Φ enforces the presence of a fixed isolated subgraph.