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] |
| Copyright of SIAM Journal on Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: COUNTING SMALL INDUCED SUBGRAPHS SATISFYING MONOTONE PROPERTIES. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22ROTH%2C+MARC%22">ROTH, MARC</searchLink><relatesTo>1</relatesTo><i> marc.roth@merton.ox.ac.uk</i><br /><searchLink fieldCode="AR" term="%22SCHMITT%2C+JOHANNES%22">SCHMITT, JOHANNES</searchLink><relatesTo>2</relatesTo><i> schmitt@math.unibonn.de</i><br /><searchLink fieldCode="AR" term="%22WELLNITZ%2C+PHILIP%22">WELLNITZ, PHILIP</searchLink><relatesTo>3</relatesTo><i> wellnitz@mpi-inf.mpg.de</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Computing%22">SIAM Journal on Computing</searchLink>. 2024, Vol. 53 Issue 6, p139-174. 36p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Linguistic+complexity%22">Linguistic complexity</searchLink><br /><searchLink fieldCode="DE" term="%22Homomorphisms%22">Homomorphisms</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Integers%22">Integers</searchLink><br /><searchLink fieldCode="DE" term="%22Logical+prediction%22">Logical prediction</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of SIAM Journal on Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1137/20M1365624 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 36 StartPage: 139 Subjects: – SubjectFull: Linguistic complexity Type: general – SubjectFull: Homomorphisms Type: general – SubjectFull: Mathematics Type: general – SubjectFull: Integers Type: general – SubjectFull: Logical prediction Type: general Titles: – TitleFull: COUNTING SMALL INDUCED SUBGRAPHS SATISFYING MONOTONE PROPERTIES. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: ROTH, MARC – PersonEntity: Name: NameFull: SCHMITT, JOHANNES – PersonEntity: Name: NameFull: WELLNITZ, PHILIP IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 11 Text: 2024 Type: published Y: 2024 Identifiers: – Type: issn-print Value: 00975397 Numbering: – Type: volume Value: 53 – Type: issue Value: 6 Titles: – TitleFull: SIAM Journal on Computing Type: main |
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