Abstract
Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area. Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of these subgraphs in some sense; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting problem. The paper also gives a thorough survey of known results on this subject and the methods used, as well as discussing the relationships both between multicolour and uncoloured versions of subgraph counting problems, and between exact counting, approximate counting and the corresponding decision problems.
Highlights
The field of parameterised counting complexity, first introduced by Flum and Grohe in [24], has received considerable attention from the theoretical computer science community in recent years; by far the most thoroughly studied family of parameterised counting problems are so-called subgraph counting problems, informally those problems which can be phrased as follows: Input: An n-vertex graph G = (V, E), and k ∈ N.Parameter: k
We extend a method based on treewidth to give some general hardness results for exact and approximate subgraph counting problems, and the related decision versions of these problems
The first approximation algorithm for a parameterised counting problem was given by Arvind and Raman in [6], and their method has recently been extended to solve a larger family of subgraph counting problems [31]
Summary
The first approximation algorithm for a parameterised counting problem was given by Arvind and Raman in [6], and their method has recently been extended to solve a larger family of subgraph counting problems [31]. Both of these positive approximation results apply to problems in which the minimal graphs that satisfy the property in question have bounded treewidth; a number of positive results for the decidability of subgraph problems exploit treewidth in a similar manner.
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