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Parameterized (Modular) Counting and Cayley Graph Expanders

Authors Norbert Peyerimhoff, Marc Roth , Johannes Schmitt , Jakob Stix, Alina Vdovina



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Author Details

Norbert Peyerimhoff
  • Department of Mathematical Sciences, Durham University, UK
Marc Roth
  • Merton College, University of Oxford, UK
Johannes Schmitt
  • Mathematical Institute, University of Bonn, Germany
Jakob Stix
  • Mathematical Institute, Goethe-Universität Frankfurt, Germany
Alina Vdovina
  • School of Mathematics and Statistics, Newcastle University, UK

Acknowledgements

The second author is grateful to Holger Dell for fruitful discussions on the connections between [Radu Curticapean et al., 2021] and our work.

Cite AsGet BibTex

Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina. Parameterized (Modular) Counting and Cayley Graph Expanders. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 84:1-84:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.84

Abstract

We study the problem #EdgeSub(Φ) of counting k-edge subgraphs satisfying a given graph property Φ in a large host graph G. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of p-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field 𝔽_p which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of #EdgeSub(Φ) for minor-closed properties Φ, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21). Additionally, we observe that our methods also apply to modular counting. Among others, we obtain novel intractability results for the problems of counting k-forests and matroid bases modulo a prime p. Furthermore, from an algorithmic point of view, we construct algorithms for the problems of counting k-paths and k-cycles modulo 2 that outperform the best known algorithms for their non-modular counterparts. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime p.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • Cayley graphs
  • counting complexity
  • expander graphs
  • fine-grained complexity
  • parameterized complexity

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References

  1. Noga Alon, Phuong Dao, Iman Hajirasouliha, Fereydoun Hormozdiari, and S. Cenk Sahinalp. Biomolecular network motif counting and discovery by color coding. Bioinformatics, 24(13):i241-i249, July 2008. URL: https://doi.org/10.1093/bioinformatics/btn163.
  2. Vikraman Arvind and Venkatesh Raman. Approximation Algorithms for Some Parameterized Counting Problems. In Algorithms and Computation, 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings, pages 453-464, 2002. URL: https://doi.org/10.1007/3-540-36136-7_40.
  3. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 684-697. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897542.
  4. Andreas Björklund, Holger Dell, and Thore Husfeldt. 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, July 6-10, 2015, Proceedings, Part I, pages 231-242, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_19.
  5. Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 151-164, 2018. URL: https://doi.org/10.1145/3188745.3188902.
  6. Andrei A. Bulatov and Amirhossein Kazeminia. Complexity classification of counting graph homomorphisms modulo a prime number. CoRR, abs/2106.04086, 2021. URL: http://arxiv.org/abs/2106.04086.
  7. Andrei A. Bulatov and Stanislav Zivný. Approximate counting CSP seen from the other side. ACM Trans. Comput. Theory, 12(2):11:1-11:19, 2020. URL: https://doi.org/10.1145/3389390.
  8. Hubie Chen and Stefan Mengel. 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, pages 315-326, 2016. URL: https://doi.org/10.1145/2902251.2902279.
  9. Radu Curticapean. Counting Matchings of Size k Is W[1]-Hard. In Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, volume 7965 of Lecture Notes in Computer Science, pages 352-363. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-39206-1_30.
  10. Radu Curticapean, Holger Dell, and Thore Husfeldt. Modular counting of subgraphs: Matchings, matching-splittable graphs, and paths. In 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal, 2021. to appear; preprint at URL: https://arxiv.org/abs/2107.00629.
  11. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 210-223, 2017. URL: https://doi.org/10.1145/3055399.3055502.
  12. 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, 2014. URL: https://doi.org/10.1109/FOCS.2014.22.
  13. Víctor Dalmau and Peter Jonsson. The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci., 329(1-3):315-323, 2004. URL: https://doi.org/10.1016/j.tcs.2004.08.008.
  14. Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 26:1-26:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.26.
  15. Arnaud Durand and Stefan Mengel. Structural Tractability of Counting of Solutions to Conjunctive Queries. Theory Comput. Syst., 57(4):1202-1249, 2015. URL: https://doi.org/10.1007/s00224-014-9543-y.
  16. John D. Faben and Mark Jerrum. The Complexity of Parity Graph Homomorphism: An Initial Investigation. Theory Comput., 11:35-57, 2015. URL: https://doi.org/10.4086/toc.2015.v011a002.
  17. Jörg Flum and Martin Grohe. The Parameterized Complexity of Counting Problems. SIAM J. Comput., 33(4):892-922, 2004. URL: https://doi.org/10.1137/S0097539703427203.
  18. Jacob Focke, Leslie Ann Goldberg, Marc Roth, and Stanislav Zivný. Counting Homomorphisms to K_4-minor-free Graphs, modulo 2. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10-13, 2021, pages 2303-2314. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.137.
  19. Fedor V. Fomin, Serge Gaspers, Saket Saurabh, and Alexey A. Stepanov. On Two Techniques of Combining Branching and Treewidth. Algorithmica, 54(2):181-207, 2009. URL: https://doi.org/10.1007/s00453-007-9133-3.
  20. Andreas Göbel, Leslie Ann Goldberg, and David Richerby. The complexity of counting homomorphisms to cactus graphs modulo 2. ACM Trans. Comput. Theory, 6(4):17:1-17:29, 2014. URL: https://doi.org/10.1145/2635825.
  21. Andreas Göbel, Leslie Ann Goldberg, and David Richerby. Counting Homomorphisms to Square-Free Graphs, Modulo 2. ACM Trans. Comput. Theory, 8(3):12:1-12:29, 2016. URL: https://doi.org/10.1145/2898441.
  22. Andreas Göbel, J. A. Gregor Lagodzinski, and Karen Seidel. Counting Homomorphisms to Trees Modulo a Prime. In Igor Potapov, Paul G. Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, volume 117 of LIPIcs, pages 49:1-49:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.49.
  23. Martin Grohe. The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM, 54(1):1:1-1:24, 2007. URL: https://doi.org/10.1145/1206035.1206036.
  24. Heng Guo, Sangxia Huang, Pinyan Lu, and Mingji Xia. The Complexity of Weighted Boolean #CSP Modulo k. In Thomas Schwentick and Christoph Dürr, editors, 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011, March 10-12, 2011, Dortmund, Germany, volume 9 of LIPIcs, pages 249-260. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2011. URL: https://doi.org/10.4230/LIPIcs.STACS.2011.249.
  25. Heng Guo, Pinyan Lu, and Leslie G. Valiant. The Complexity of Symmetric Boolean Parity Holant Problems. SIAM J. Comput., 42(1):324-356, 2013. URL: https://doi.org/10.1137/100815530.
  26. Amirhossein Kazeminia and Andrei A. Bulatov. Counting Homomorphisms Modulo a Prime Number. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 59:1-59:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.59.
  27. J. A. Gregor Lagodzinski, Andreas Göbel, Katrin Casel, and Tobias Friedrich. On Counting (Quantum-)Graph Homomorphisms in Finite Fields. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, to appear, 2021. Preprint on URL: https://arxiv.org/abs/2011.04827.
  28. Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Efficient Computation of Representative Weight Functions with Applications to Parameterized Counting. Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Alexandria, VA, USA, January 10-13, 2021, to appear. Google Scholar
  29. László Lovász. Large Networks and Graph Limits, volume 60 of Colloquium Publications. American Mathematical Society, 2012. URL: http://www.ams.org/bookstore-getitem/item=COLL-60.
  30. A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. URL: https://doi.org/10.1007/BF02126799.
  31. Dániel Marx. Can You Beat Treewidth? Theory of Computing, 6(1):85-112, 2010. URL: https://doi.org/10.4086/toc.2010.v006a005.
  32. Dániel Marx. Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries. J. ACM, 60(6):42:1-42:51, 2013. URL: https://doi.org/10.1145/2535926.
  33. Catherine McCartin. Parameterized counting problems. Ann. Pure Appl. Logic, 138(1-3):147-182, 2006. URL: https://doi.org/10.1016/j.apal.2005.06.010.
  34. R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. 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.
  35. Moshe Morgenstern. Existence and explicit constructions of q+1 regular Ramanujan graphs for every prime power q. J. Combin. Theory Ser. B, 62(1):44-62, 1994. URL: https://doi.org/10.1006/jctb.1994.1054.
  36. Norbert Peyerimhoff and Alina Vdovina. Cayley graph expanders and groups of finite width. J. Pure Appl. Algebra, 215(11):2780-2788, 2011. URL: https://doi.org/10.1016/j.jpaa.2011.03.018.
  37. 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.
  38. Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting Small Induced Subgraphs Satisfying Monotone Properties. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1356-1367. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00128.
  39. 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 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, to appear, 2021. Preprint on URL: https://arxiv.org/abs/2011.03433.
  40. Nithi Rungtanapirom, Jakob Stix, and Alina Vdovina. Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical ramanujan complexes. International Journal of Algebra and Computation, 29(6):951-1007, 2019. Google Scholar
  41. 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, pages 53-67, Cham, 2015. Springer International Publishing. Google Scholar
  42. Jakob Stix and Alina Vdovina. Simply transitive quaternionic lattices of rank 2 over 𝔽_q(t) and a non-classical fake quadric. Mathematical Proceedings of the Cambridge Philosophical Society, 163(3):453-498, 2017. Google Scholar
  43. Seinosuke Toda. PP is as Hard as the Polynomial-Time Hierarchy. SIAM J. Comput., 20(5):865-877, 1991. URL: https://doi.org/10.1137/0220053.
  44. Leslie G. Valiant. The Complexity of Computing the Permanent. Theor. Comput. Sci., 8:189-201, 1979. URL: https://doi.org/10.1016/0304-3975(79)90044-6.
  45. Leslie G. Valiant. Holographic Algorithms. SIAM J. Comput., 37(5):1565-1594, 2008. URL: https://doi.org/10.1137/070682575.
  46. Virginia Vassilevska Williams, Joshua R. Wang, Richard Ryan Williams, and Huacheng Yu. 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, January 4-6, 2015, pages 1671-1680, 2015. URL: https://doi.org/10.1137/1.9781611973730.111.
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