Some hard families of parameterised counting problems

Mark Jerrum; Kitty Meeks

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arXiv · 1310.6524

Some hard families of parameterised counting problems

Mark Jerrum·Kitty Meeks

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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.

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Mark Jerrum, Kitty Meeks. 2014-09-25. Some hard families of parameterised counting problems. https://arxiv.org/abs/1310.6524

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