COUNTING SMALL INDUCED SUBGRAPHS SATISFYING MONOTONE PROPERTIES.
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| Title: | COUNTING SMALL INDUCED SUBGRAPHS SATISFYING MONOTONE PROPERTIES. |
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| Authors: | ROTH, MARC1 marc.roth@merton.ox.ac.uk, SCHMITT, JOHANNES2 schmitt@math.unibonn.de, WELLNITZ, PHILIP3 wellnitz@mpi-inf.mpg.de |
| Source: | SIAM Journal on Computing. 2024, Vol. 53 Issue 6, p139-174. 36p. |
| Subjects: | Linguistic complexity, Homomorphisms, Mathematics, Integers, Logical prediction |
| Abstract: | Given a graph property \Phi, the problem \#IndSub(\Phi) asks, on input of a graph G and a positive integer k, to compute the number \#\sansI \sansn \sansd \sansS \sansu \sansb (\Phi, k \rightarrow G) of induced subgraphs of size k in G that satisfy \Phi. The search for explicit criteria on \Phi ensuring that \#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 D\"orfler 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 \#IndSub(\Phi) cannot be solved in time f(k)\cdot | V (G)| o(k/log1/2(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 \#\sansW [\sansone ]-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]: \#IndSub(\Phi) is \#\sansW [\sansone ]-complete if \Phi is a nontrivial graph property only depending on the number of edges of the graph. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
| Abstract: | Given a graph property \Phi, the problem \#IndSub(\Phi) asks, on input of a graph G and a positive integer k, to compute the number \#\sansI \sansn \sansd \sansS \sansu \sansb (\Phi, k \rightarrow G) of induced subgraphs of size k in G that satisfy \Phi. The search for explicit criteria on \Phi ensuring that \#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 D\"orfler 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 \#IndSub(\Phi) cannot be solved in time f(k)\cdot | V (G)| o(k/log1/2(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 \#\sansW [\sansone ]-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]: \#IndSub(\Phi) is \#\sansW [\sansone ]-complete if \Phi is a nontrivial graph property only depending on the number of edges of the graph. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 00975397 |
| DOI: | 10.1137/20M1365624 |
