Some hard families of parameterised counting problems
Abstract
We consider parameterised subgraph-counting problems of the following form: given a graph G, how many k-tuples of its vertices have a given property? A number of such problems are known to be #W[1]-complete; here we substantially generalise 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 labelled subgraphs that satisfy the property is o(k^2). In the special case that 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.
Cite
@article{arxiv.1310.6524,
title = {Some hard families of parameterised counting problems},
author = {Mark Jerrum and Kitty Meeks},
journal= {arXiv preprint arXiv:1310.6524},
year = {2014}
}
Comments
A few more minor changes. This version to appear in the ACM Transactions on Computation Theory