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Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

Authors Radu Curticapean , Simon Döring , Daniel Neuen

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  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
Keywords
  • induced subgraphs
  • counting complexity
  • parameterized complexity
  • scorpions

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Abstract

In the parameterized problem #IndSub(Φ) for fixed graph properties Φ, given as input a graph G and an integer k, the task is to compute the number of induced k-vertex subgraphs satisfying Φ. Dörfler et al. [Algorithmica 2022] and Roth et al. [SICOMP 2024] conjectured that #IndSub(Φ) is #W[1]-hard for all non-meager properties Φ, i.e., properties that are nontrivial for infinitely many k. This conjecture has been confirmed for several restricted types of properties, including all hereditary properties [STOC 2022] and all edge-monotone properties [STOC 2024].
We refute this conjecture by showing that induced k-vertex graphs that are scorpions can be counted in time O(n⁴) for all k. Scorpions were introduced more than 50 years ago in the context of the evasiveness conjecture. A simple variant of this construction results in graph properties that achieve arbitrary intermediate complexity assuming ETH.
Moreover, we formulate an updated conjecture on the complexity of #IndSub(Φ) that correctly captures the complexity status of scorpions and related constructions.

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Radu Curticapean, Simon Döring, and Daniel Neuen. Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 96:1-96:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.96

Author Details

Radu Curticapean
  • University of Regensburg, Germany
  • IT University of Copenhagen, Denmark
Simon Döring
  • Max Planck Institute for Informatics, Saarbrücken, Germany
  • Saarland University (SIC), Saarbrücken, Germany
Daniel Neuen
  • Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

The research is funded by the European Union (ERC, CountHom, 101077083). Views and opinions expressed are those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

References

  1. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More asymmetry yields faster matrix multiplication. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 2005-2039. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.63.
  2. Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Counting graph homomorphisms. In Topics in discrete mathematics, volume 26 of Algorithms Combin., pages 315-371. Springer, Berlin, 2006. URL: https://doi.org/10.1007/3-540-33700-8_18.
  3. Karl Bringmann and Jasper Slusallek. Current algorithms for detecting subgraphs of bounded treewidth are probably optimal. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 40:1-40:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.40.
  4. Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci., 72(8):1346-1367, 2006. URL: https://doi.org/10.1016/J.JCSS.2006.04.007.
  5. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 210-223. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055502.
  6. Radu Curticapean, Simon Döring, Daniel Neuen, and Jiaheng Wang. Can you link up with treewidth? In Olaf Beyersdorff, Michal Pilipczuk, Elaine Pimentel, and Kim Thang Nguyen, editors, 42nd International Symposium on Theoretical Aspects of Computer Science, STACS 2025, March 4-7, 2025, Jena, Germany, volume 327 of LIPIcs, pages 28:1-28:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPICS.STACS.2025.28.
  7. Radu Curticapean and Dániel Marx. Complexity of counting subgraphs: Only the boundedness of the vertex-cover number counts. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 130-139. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.22.
  8. Radu Curticapean and Daniel Neuen. Counting small induced subgraphs: Hardness via Fourier analysis. CoRR, abs/2407.07051, 2024. URL: https://doi.org/10.48550/arXiv.2407.07051.
  9. Radu Curticapean and Daniel Neuen. Counting small induced subgraphs: Hardness via Fourier analysis. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 3677-3695. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.122.
  10. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  11. Reinhard Diestel. Graph Theory. Springer Berlin, 5 edition, 2017. URL: https://doi.org/10.1007/978-3-662-53622-3.
  12. Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting induced subgraphs: An algebraic approach to #W[1]-hardness. Algorithmica, 84(2):379-404, 2022. URL: https://doi.org/10.1007/S00453-021-00894-9.
  13. Simon Döring, Dániel Marx, and Philip Wellnitz. Counting small induced subgraphs with edge-monotone properties. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 1517-1525. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649644.
  14. Simon Döring, Dániel Marx, and Philip Wellnitz. From graph properties to graph parameters: Tight bounds for counting on small subgraphs. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 3637-3676. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.121.
  15. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  16. Jacob Focke and Marc Roth. Counting small induced subgraphs with hereditary properties. SIAM J. Comput., 53(2):189-220, 2024. URL: https://doi.org/10.1137/22M1512211.
  17. Leslie Ann Goldberg and Marc Roth. Parameterised and fine-grained subgraph counting, modulo 2. Algorithmica, 86(4):944-1005, 2024. URL: https://doi.org/10.1007/S00453-023-01178-0.
  18. Martin Grohe, Thomas Schwentick, and Luc Segoufin. When is the evaluation of conjunctive queries tractable? In Jeffrey Scott Vitter, Paul G. Spirakis, and Mihalis Yannakakis, editors, Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 657-666. ACM, 2001. URL: https://doi.org/10.1145/380752.380867.
  19. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/JCSS.2000.1727.
  20. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/JCSS.2001.1774.
  21. Alon Itai and Michael Rodeh. Finding a minimum circuit in a graph. SIAM J. Comput., 7(4):413-423, 1978. URL: https://doi.org/10.1137/0207033.
  22. Mark Jerrum and Kitty Meeks. The parameterised complexity of counting connected subgraphs and graph motifs. J. Comput. Syst. Sci., 81(4):702-716, 2015. URL: https://doi.org/10.1016/J.JCSS.2014.11.015.
  23. Mark Jerrum and Kitty Meeks. Some hard families of parameterized counting problems. ACM Trans. Comput. Theory, 7(3):11:1-11:18, 2015. URL: https://doi.org/10.1145/2786017.
  24. Emily Jin, Michael M. Bronstein, İsmail İlkan Ceylan, and Matthias Lanzinger. Homomorphism counts for graph neural networks: All about that basis. In Forty-first International Conference on Machine Learning, ICML 2024, Vienna, Austria, July 21-27, 2024. OpenReview.net, 2024. URL: https://openreview.net/forum?id=zRrzSLwNHQ.
  25. Jeff Kahn, Michael E. Saks, and Dean Sturtevant. A topological approach to evasiveness. Comb., 4(4):297-306, 1984. URL: https://doi.org/10.1007/BF02579140.
  26. Dmitry N. Kozlov. Combinatorial Algebraic Topology, volume 21 of Algorithms and computation in mathematics. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-71962-5.
  27. Hendrik Lenstra, Marc R. Best, and Peter van Emde Boas. A sharpened version of the Aanderaa-Rosenberg conjecture. Report 30/74, Mathematisch Centrum Amsterdam (1974), pages 1-20, 1974. URL: https://hdl.handle.net/1887/3792.
  28. Ron Milo, Shai S. Shen-Orr, Shalev Itzkovitz, Nadav Kashtan, Dmitri B. Chklovskii, and Uri Alon. Network motifs: Simple building blocks of complex networks. Science, 298(5594):824-827, 2002. URL: https://doi.org/10.1126/science.298.5594.824.
  29. Peter M. Neumann. Transitive permutation groups of prime degree. In Proceedings of the Second International Conference on the Theory of Groups, volume Vol. 372 of Lecture Notes in Math., pages 520-535. Springer Berlin Heidelberg, 1974. URL: https://doi.org/10.1007/978-3-662-21571-5_55.
  30. Jaroslav Nešetřil and Svatopluk Poljak. On the complexity of the subgraph problem. Comment. Math. Univ. Carolin., 26(2):415-419, 1985. URL: http://dml.cz/dmlcz/106381.
  31. Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. Comput. Complex., 4:301-313, 1994. URL: https://doi.org/10.1007/BF01263419.
  32. Ronald L. Rivest and Jean Vuillemin. On recognizing graph properties from adjacency matrices. Theor. Comput. Sci., 3(3):371-384, 1976. URL: https://doi.org/10.1016/0304-3975(76)90053-0.
  33. Arnold L. Rosenberg. On the time required to recognize properties of graphs: a problem. SIGACT News, 5(4):15-16, 1973. URL: https://doi.org/10.1145/1008299.1008302.
  34. Moshe Rosenfeld. Independent sets in regular graphs. Israel J. Math., 2:262-272, 1964. URL: https://doi.org/10.1007/BF02759743.
  35. Marc Roth and Johannes Schmitt. Counting induced subgraphs: A topological approach to #W[1]-hardness. Algorithmica, 82(8):2267-2291, 2020. URL: https://doi.org/10.1007/S00453-020-00676-9.
  36. Marc Roth, Johannes Schmitt, and Philip Wellnitz. Detecting and counting small subgraphs, and evaluating a parameterized tutte polynomial: Lower bounds via toroidal grids and cayley graph expanders. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 108:1-108:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.108.
  37. Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting small induced subgraphs satisfying monotone properties. SIAM J. Comput., 53(6):FOCS20-139-FOCS20-174, 2024. URL: https://doi.org/10.1137/20M1365624.
  38. Benjamin Schiller, Sven Jager, Kay Hamacher, and Thorsten Strufe. StreaM - A stream-based algorithm for counting motifs in dynamic graphs. In Adrian-Horia Dediu, Francisco Hernández Quiroz, Carlos Martín-Vide, and David A. Rosenblueth, editors, Algorithms for Computational Biology - Second International Conference, AlCoB 2015, Mexico City, Mexico, August 4-5, 2015, Proceedings, volume 9199 of Lecture Notes in Computer Science, pages 53-67. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21233-3_5.
  39. Falk Schreiber and Henning Schwöbbermeyer. Frequency concepts and pattern detection for the analysis of motifs in networks. Trans. Comp. Sys. Biology, 3:89-104, 2005. URL: https://doi.org/10.1007/11599128_7.
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