Some Hard Families of Parameterized Counting Problems

摘要

We consider parameterized subgraph counting problems of the following form: given a graph G, how many k-tuples of its vertices induce a subgraph with a given property? A number of such problems are known to be #W[1]-complete; here, we substantially generalize some of these existing results by proving hardness for two large families of such problems. We demonstrate that it is #W[1]-hard to count the number of k-vertex subgraphs having any property where the number of distinct edge densities of labeled subgraphs that satisfy the property is o(k~2). In the special case in which the property in question depends only on the number of edges in the subgraph, we give a strengthening of this result, which leads to our second family of hard problems.