Abstract
We investigate the problem #{{mathsf {IndSub}}}(varPhi ) of counting all induced subgraphs of size k in a graph G that satisfy a given property varPhi .This continues the work of Jerrum and Meeks who proved the problem to be #{{mathrm {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 , the problem #{{mathsf {IndSub}}}(varPhi ) is hard for #{{mathrm {W[1]}}} if the reduced Euler characteristic of the associated simplicial (graph) complex of varPhi is non-zero. This observation links #{{mathsf {IndSub}}}(varPhi ) 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 ) is #{{mathrm {W[1]}}}-hard for every monotone property varPhi 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 #{{mathsf {IndSub}}}(varPhi ) can not be solved in time f(k)cdot 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 ) is #{{mathrm {W[1]}}}-hard if varPhi is any non-trivial modularity constraint on the number of edges with respect to some prime q or if varPhi enforces the presence of a fixed isolated subgraph.
Highlights
In their work about the parameterized complexity of counting problems [11] Flum and Grohe introduced the parameterized analogue of the theory of computational counting as layed out by Valiant in his seminal paper about the complexity of computing the permanent [29]
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
It turns out that for mono∑tone properties, i.e., properties that are closed under the removal of edges, the term A∈ k (−1)#A is equal to the reduced Euler characteristic χof the simplicial graph complex of k
Summary
In their work about the parameterized complexity of counting problems [11] Flum and Grohe introduced the parameterized analogue of the theory of computational counting as layed out by Valiant in his seminal paper about the complexity of computing the permanent [29]. It turned out that many families of such counting problems allow so-called dichotomy results, that is, every problem in the family is either fixed-parameter tractable or hard for the class #W[1]—the counting equivalent of W[1]. One result of that kind is the dichotomy for counting homomorphisms [10, 13]. One is given a graph H from a class of graphs H and an arbitrary graph G and the task is to compute the number of homomorphisms from H to G
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