Counting Induced Subgraphs: An Algebraic Approach to #W[1]-Hardness

We study the problem # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property Φ . It is shown that, given any graph property Φ that distinguishes independent sets from bicliques, # I N D S U B ( Φ ) is hard for the class # W [ 1 ] , i.e., the parameteriz...

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Veröffentlicht in:	Algorithmica Jg. 84; H. 2; S. 379 - 404
Hauptverfasser:	Dörfler, Julian, Roth, Marc, Schmitt, Johannes, Wellnitz, Philip
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.02.2022 Springer Nature B.V
Schlagworte:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Hardness Mathematics of Computing Theory of Computation Counting complexity Edge-transitive graphs Graph homomorphisms Induced subgraphs Parameterized complexity
ISSN:	0178-4617, 1432-0541
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Abstract	We study the problem # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property Φ . It is shown that, given any graph property Φ that distinguishes independent sets from bicliques, # I N D S U B ( Φ ) is hard for the class # W [ 1 ] , i.e., the parameterized counting equivalent of N P . Under additional suitable density conditions on Φ , satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen # W [ 1 ] -hardness by establishing that # I N D S U B ( Φ ) cannot be solved in time f ( k ) · n o ( k ) for any computable function f , unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime.
AbstractList	We study the problem $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property $$\varPhi $$ Φ . It is shown that, given any graph property $$\varPhi $$ Φ that distinguishes independent sets from bicliques, $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) is hard for the class $$\#\mathsf {W[1]}$$ # W [ 1 ] , i.e., the parameterized counting equivalent of $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P . Under additional suitable density conditions on $$\varPhi $$ Φ , satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen $$\#\mathsf {W[1]}$$ # W [ 1 ] -hardness by establishing that $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) cannot be solved in time $$f(k)\cdot n^{o(k)}$$ f ( k ) · n o ( k ) for any computable function f , unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime. We study the problem #INDSUB(Φ) of counting all induced subgraphs of size k in a graph G that satisfy the property Φ. It is shown that, given any graph property Φ that distinguishes independent sets from bicliques, #INDSUB(Φ) is hard for the class #W[1], i.e., the parameterized counting equivalent of NP. Under additional suitable density conditions on Φ, satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen #W[1]-hardness by establishing that #INDSUB(Φ) cannot be solved in time f(k)·no(k) for any computable function f, unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime. We study the problem # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property Φ . It is shown that, given any graph property Φ that distinguishes independent sets from bicliques, # I N D S U B ( Φ ) is hard for the class # W [ 1 ] , i.e., the parameterized counting equivalent of N P . Under additional suitable density conditions on Φ , satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen # W [ 1 ] -hardness by establishing that # I N D S U B ( Φ ) cannot be solved in time f ( k ) · n o ( k ) for any computable function f , unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime.
Author	Dörfler, Julian Schmitt, Johannes Roth, Marc Wellnitz, Philip
Author_xml	– sequence: 1 givenname: Julian surname: Dörfler fullname: Dörfler, Julian organization: Graduate School of Computer Science, Saarland Informatics Campus – sequence: 2 givenname: Marc orcidid: 0000-0003-3159-9418 surname: Roth fullname: Roth, Marc email: marc.roth@cs.ox.ac.uk organization: Department of Computer Science and Merton College, University of Oxford, Cluster of Excellence (MMCI), Saarland Informatics Campus – sequence: 3 givenname: Johannes surname: Schmitt fullname: Schmitt, Johannes organization: ETH Zurich, Mathematical Institute, University of Bonn – sequence: 4 givenname: Philip surname: Wellnitz fullname: Wellnitz, Philip organization: Max Planck Institute for Informatics, Saarland Informatics Campus
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Cites_doi	10.1016/0304-3975(79)90044-6 10.2307/2033425 10.1016/j.dam.2015.06.019 10.4153/CJM-1965-045-4 10.1142/9789812773449 10.1090/coll/060 10.1007/BF02579140 10.1145/2902251.2902279 10.1016/j.tcs.2004.08.008 10.1137/1.9781611973730.111 10.1145/2786017 10.1016/j.ic.2005.05.001 10.1007/978-3-662-47672-7_19 10.1007/3-540-29953-X 10.1145/3055399.3055502 10.1016/j.apal.2005.06.010 10.4230/LIPIcs.IPEC.2018.24 10.1137/S0097539703427203 10.1137/0220053 10.1016/0304-3975(76)90053-0 10.1016/j.jcss.2006.04.007 10.1007/s00493-016-3338-5 10.1017/CBO9780511608704 10.1016/j.jcss.2014.11.015 10.1561/0400000055
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Keywords	Counting complexity Edge-transitive graphs Graph homomorphisms Induced subgraphs Parameterized complexity
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Cambridge University Press, Cambridge (1993). https://doi.org/10.1017/CBO9780511608704 ChenJHuangXKanjIAXiaGStrong computational lower bounds via parameterized complexityJ. Comput. Syst. Sci.200672813461367227341210.1016/j.jcss.2006.04.0071119.68092 MeeksKThe challenges of unbounded treewidth in parameterised subgraph counting problemsDiscrete Appl. Math.2016198170194342689010.1016/j.dam.2015.06.0191327.05244 Curticapean, R.: The simple, little and slow things count: On parameterized counting complexity. Ph.D. thesis, Saarland University (2015). http://scidok.sulb.uni-saarland.de/volltexte/2015/6217 JerrumMMeeksKThe parameterised complexity of counting even and odd induced subgraphsCombinatorica2017375965990373737610.1007/s00493-016-3338-51413.68063 TodaSPP is as hard as the polynomial-time hierarchySIAM J. Comput.1991205865877111565510.1137/02200530733.68034 Lovász, L.: Large networks and graph limits, Colloquium Publications, vol. 60. 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References_xml	– reference: McCartinCParameterized counting problemsAnn. Pure Appl. Logic20061381–3147182218381210.1016/j.apal.2005.06.0101133.68027 – reference: MeeksKThe challenges of unbounded treewidth in parameterised subgraph counting problemsDiscrete Appl. Math.2016198170194342689010.1016/j.dam.2015.06.0191327.05244 – reference: Curticapean, R.: The simple, little and slow things count: On parameterized counting complexity. Ph.D. thesis, Saarland University (2015). http://scidok.sulb.uni-saarland.de/volltexte/2015/6217/ – reference: Chen, H., Mengel, S.: Counting answers to existential positive queries: A complexity classification. In: Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2016, San Francisco, CA, USA, June 26–July 01, 2016, pp. 315–326 (2016). https://doi.org/10.1145/2902251.2902279 – reference: EdmondsJPaths, trees, and flowersCan. J. Math.19651744946717790710.4153/CJM-1965-045-40132.20903 – reference: JerrumMMeeksKThe parameterised complexity of counting even and odd induced subgraphsCombinatorica2017375965990373737610.1007/s00493-016-3338-51413.68063 – reference: TodaSPP is as hard as the polynomial-time hierarchySIAM J. Comput.1991205865877111565510.1137/02200530733.68034 – reference: Björklund, A., Dell, H., Husfeldt, T.: The parity of set systems under random restrictions with applications to exponential time problems. In: Automata, Languages, and Programming—42nd International Colloquium, ICALP 2015, Kyoto, Japan, 6–10 July, 2015, Proceedings, Part I, pp. 231–242 (2015). https://doi.org/10.1007/978-3-662-47672-7_19 – reference: FlumJGroheMThe parameterized complexity of counting problemsSIAM J. Comput.2004334892922206533810.1137/S00975397034272031105.68042 – reference: ChenJChorBFellowsMHuangXJuedesDWKanjIAXiaGTight lower bounds for certain parameterized NP-hard problemsInf. Comput.20052012216231216571610.1016/j.ic.2005.05.0011161.68476 – reference: WeirAJThe Sylow subgroups of the symmetric groupsProc. Amer. Math. Soc.195565345417214210.2307/20334250065.25602 – reference: Williams, V.V., Wang, J.R., Williams, R.R., Yu, H.: Finding four-node subgraphs in triangle time. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, 4–6 January, 2015, pp. 1671–1680 (2015). https://doi.org/10.1137/1.9781611973730.111 – reference: Biggs, N.: Algebraic graph theory, 2nd (edn.). Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993). https://doi.org/10.1017/CBO9780511608704 – reference: Roth, M., Schmitt, J.: Counting induced subgraphs: A topological approach to #W[1]-hardness. In: 13th International Symposium on Parameterized and Exact Computation, IPEC 2018, 20–24 August, 2018, Helsinki, Finland, pp. 24:1–24:14 (2018). https://doi.org/10.4230/LIPIcs.IPEC.2018.24 – reference: Abhyankar, S.S.: Lectures on algebra, vol. I. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). https://doi.org/10.1142/9789812773449 – reference: DalmauVJonssonPThe complexity of counting homomorphisms seen from the other sideTheor. Comput. Sci.20043291–3315323210365510.1016/j.tcs.2004.08.0081086.68054 – reference: ChenJHuangXKanjIAXiaGStrong computational lower bounds via parameterized complexityJ. Comput. Syst. Sci.200672813461367227341210.1016/j.jcss.2006.04.0071119.68092 – reference: KahnJSaksMESturtevantDA topological approach to evasivenessCombinatorica19844429730677989010.1007/BF025791400577.05061 – reference: Flum, J., Grohe, M.: Parameterized complexity theory. Texts in Theoretical Computer Science. An EATCS Series. 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Snippet	We study the problem # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property Φ . It is shown that, given any... We study the problem $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property... We study the problem #INDSUB(Φ) of counting all induced subgraphs of size k in a graph G that satisfy the property Φ. It is shown that, given any graph...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Hardness Mathematics of Computing Theory of Computation
Title	Counting Induced Subgraphs: An Algebraic Approach to #W[1]-Hardness
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