Counting Small Induced Subgraphs Satisfying Monotone Properties

Abstract

Given a graph property $\Phi$, the problem $\#\ensuremath{{\sc IndSub}}(\Phi)$ asks, on input of a graph $G$ and a positive integer $k$, to compute the number $\#{\ensuremath{{IndSub}({\Phi,k} \to {G})}}$ of induced subgraphs of size $k$ in $G$ that satisfy $\Phi$. The search for explicit criteria on $\Phi$ ensuring that $\#\ensuremath{{\sc IndSub}}(\Phi)$ is hard was initiated by Jerrum and Meeks [J. Comput. System Sci., 81 (2015), pp. 702--716] 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, ACM, New York, pp. 151--158] proving that a full classification into “easy” and “hard” properties is possible and some partial results on edge-monotone properties due to Meeks [Discrete Appl. Math., 198 (2016), pp. 170--194] and Dörfler et al. [MFCS, LIPIcs Leibniz Int. Proc. Inform. 138, Wadern Germany, 2019, 26], 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 nontrivial monotone property $\Phi$, the problem $\#\ensuremath{{\sc IndSub}}(\Phi)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ 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 $\#{\ensuremath{{W[1]}}}$-completeness result. The methods we develop for the above problem also allow us to prove a conjecture by Jerrum and Meeks [ACM Trans. Comput. Theory, 7 (2015), 11; Combinatorica 37 (2017), pp. 965--990]: $\#\ensuremath{{\sc IndSub}}(\Phi)$ is $\#{\ensuremath{{W[1]}}}$-complete if $\Phi$ is a nontrivial graph property only depending on the number of edges of the graph.