Counting Small Induced Subgraphs with Hereditary Properties

Abstract

We study the computational complexity of the problem #IndSub(Φ) of counting k-vertex induced subgraphs of a graph G that satisfy a graph property Φ. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH):• If a hereditary property Φ is true for all graphs, or if it is true only for finitely many graphs, then #IndSub(Φ) is solvable in polynomial time.• Otherwise, #IndSub(Φ) is #W[1]-complete when parameterised by k, and, assuming ETH, it cannot be solved in timeThis classification features a wide range of properties for which the corresponding detection problem (as classified by Khot and Raman [TCS 02]) is tractable but counting is hard. Moreover, even for properties which are already intractable in their decision version, our results yield significantly stronger lower bounds for the counting problem.As additional result, we also present an exhaustive and explicit parameterised complexity classification for all properties that are invariant under homomorphic equivalence.By covering one of the most natural and general notions of closure, namely, closure under vertexdeletion (hereditary), we generalise some of the earlier results on this problem. For instance, our results fully subsume and strengthen the existing classification of #IndSub(Φ) for monotone (subgraph-closed) properties due to Roth, Schmitt, and Wellnitz [FOCS 20].

Counting Small Induced Subgraphs with Hereditary Properties

“…In addition to confirming Conjecture 1 for hereditary properties, we also establish a tight conditional lower bound under the Exponential Time Hypothesis; it turns out that a hereditary property is meagre if and only if either Φ is true for all graphs, or it is true only for finitely many graphs (an easy proof of this fact can be found in the full version [15]). Observe that our conditional lower bound under ETH rules out any significant improvement over the brute-force algorithm for #IndSub(Φ), which iterates over every 𝑘-vertex subset of 𝑉 (𝐺) and counts those that induce a subgraph satisfying Φ.…”

“…For example, the property Φ of having an even number of vertices is meagre, and it is easy to see that #IndSub(Φ) is trivial to solve: On input 𝐺 and 𝑘, output 0 if 𝑘 is odd, and output |𝑉 (𝐺) | 𝑘 if 𝑘 is even. It is well-known that an analogue of the previous algorithm exists for every meagre property; this is made formal in the full version [15]. Conversely, as stated in [30], we conjecture that all non-meagre properties yield hardness:…”

Counting small induced subgraphs with hereditary properties

“…In addition to confirming Conjecture 1 for hereditary properties, we also establish a tight conditional lower bound under the Exponential Time Hypothesis; it turns out that a hereditary property is meagre if and only if either Φ is true for all graphs, or it is true only for finitely many graphs (an easy proof of this fact can be found in the full version [15]). Theorem 2.…”

“…In the course of establishing Theorem 2, we prove a much stronger technical intractability theorem which is stated in the full version [15]. We have not yet explored the full extent of its applicability and we believe that it will be useful in future work (see Section 5).…”

“…In[19], #IndSub(Φ) is called #InducedUnlabelledSubgraphWithProperty(Φ) 2. The class #W[1] can be considered as a parameterised counting equivalent of NP; a formal definition is stated in the full version[15].…”