Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property Φ, the problem #IndSub(Φ) asks, on input a graph G and a positive integer k, to compute the number #IndSub(Φ, k → G) of induced subgraphs of size k in G that satisfy Φ. The search for explicit criteria on Φ ensuring that #IndSub(Φ) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and Dörfler et al. [MFCS 19], not much is known.

In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property Φ, the problem #IndSub(Φ) cannot be solved in time

for any function f , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a #W[1]-completeness result.

To prove our result, we use that for fixed Φ and k, we can express the function G → #IndSub(Φ, k → G) as a finite linear-combination of homomorphism counts from graphs Hi to G. The coefficient vectors of these homomorphism counts in the linear combination are called the homomorphism vectors associated to Φ; by the Complexity Monotonicity framework of Curticapean, Dell and Marx [STOC 17], the positions of non-zero entries of these vectors are known to determine the complexity of #IndSub(Φ). Our main technical result lifts the notion of f -polynomials from simplicial complexes to graph properties and relates the derivatives of the f -polynomial of Φ to its homomorphism vector. We then apply results from the theory of Hermite-Birkhoff interpolation to the f -polynomial to establish sufficient conditions on Φ which ensure that certain entries in the homomorphism vector do not vanish-which in turn implies hardness. For monotone graph properties, non-triviality then turns out to be a sufficient condition. Using the same method, we also prove a conjecture by Jerrum and Meeks [TOCT 15, Combinatorica 19]: #IndSub(Φ) is #W[1]-complete if Φ is a non-trivial graph property only depending on the number of edges of the graph.