Counting Small Induced Subgraphs Satisfying Monotone Properties
Abstract
Given a graph property \(\Phi\), the problem \(\#\text{I}{\small{\text{ND}}}\text{S}{\small{\text{UB}}}(\Phi)\) asks, on input of a graph \(G\) and a positive integer \(k\), to compute the number \(\#\textsf{IndSub}(\Phi,k\rightarrow G)\) of induced subgraphs of size \(k\) in \(G\) that satisfy \(\Phi\). The search for explicit criteria on \(\Phi\) ensuring that \(\#\text{I}{\small{\text{ND}}}\text{S}{\small{\text{UB}}}(\Phi)\) 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 \(\#\text{I}{\small{\text{ND}}}\text{S}{\small{\text{UB}}}(\Phi)\) cannot be solved in time \(f(k)\cdot \vert V(G)\vert^{o(k/\log^{1/2}(k))}\) 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 \(\#\textsf{W}[1]\)-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]: \(\#\text{I}{\small{\text{ND}}}\text{S}{\small{\text{UB}}}(\Phi)\) is \(\#\textsf{W}[1]\)-complete if \(\Phi\) is a nontrivial graph property only depending on the number of edges of the graph.
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SIAM Journal on Computing
Pages: FOCS20-139 - FOCS20-174
DOI: 10.1137/20M1365624
ISSN (online): 1095-7111
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Submitted: 8 September 2020
Accepted: 3 November 2021
Published online: 11 April 2022
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