Counting Small Induced Subgraphs Satisfying Monotone Properties
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Given a graph property Φ, the problem #𝖨𝗇𝖽𝖲𝗎𝖻(Φ) asks, on input a graph G and a positive integer k, to compute the number 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 f(k)· |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 #𝖶[1]-completeness result.
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