The challenges of unbounded treewidth in parameterised subgraph counting problems

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.

Randomised Enumeration of Small Witnesses Using a Decision Oracle

“…This complements the fact that a simple random sampling algorithm can be used for approximate counting when the number of witnesses is very large [22,Lemma 3.4], although there remain many situations which are not covered by either result.…”

“…However, in the case that the number N of witnesses is large, an enumeration algorithm necessarily takes time at least (N ), whereas we might hope for much better if our goal is simply to determine the total number of witnesses. The family of self-contained k-witness problems studied here includes subgraph problems, whose parameterised complexity from the point of view of counting has been a rich topic for research in recent years [10,11,14,[17][18][19]22]. Many such counting problems, including those whose decision problem belongs to FPT, are known to be #W [1]-complete (see [15] for background on the theory of parameterised counting complexity).…”

Homomorphisms are a good basis for counting small subgraphs

“…Let us also address the problem of counting small induced subgraphs from a class H. This is a natural and well-studied variant of counting subgraph copies [38,13,33,34,35,45], and for several applications it represents a more appropriate notion of "pattern occurrence". From the perspective of dichotomy results however, it is less intricate than subgraphs or homomorphisms: Counting induced subgraphs is known to be #W [1] [33,34,35,45] introduced the following generalization of the problems #Ind(H) to fixed graph properties Φ: Given a graph G and k ∈ N, the task is to compute the number of induced k-vertex subgraphs that have property Φ.…”

“…From the perspective of dichotomy results however, it is less intricate than subgraphs or homomorphisms: Counting induced subgraphs is known to be #W [1] [33,34,35,45] introduced the following generalization of the problems #Ind(H) to fixed graph properties Φ: Given a graph G and k ∈ N, the task is to compute the number of induced k-vertex subgraphs that have property Φ. Let us call this problem #IndProp(Φ).…”

Engineering a Delegatable and Error-Tolerant Algorithm for Counting Small Subgraphs

“…Cf. Flum and Grohe [50], Chen and Flum [34], Chen, Thurley, Weyer [35], Curticapean [39], Curticapean and Marx [41], Jerrum and Meeks [69,70], and Meeks [81]. The specific problem of finding and counting cliques is used as a source of fine-grained hardness reductions by Abboud, Backurs, and Vassilevska Williams [1].…”