Document Open Access Logo

A Dichotomy Theorem for Linear Time Homomorphism Orbit Counting in Bounded Degeneracy Graphs

Authors Daniel Paul-Pena , C. Seshadhri

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2024.54.pdf
  • Filesize: 0.8 MB
  • 19 pages

Document Identifiers

Author Details

Daniel Paul-Pena
  • University of California, Santa Cruz, CA, USA
C. Seshadhri
  • University of California, Santa Cruz, CA, USA

Funding

Both authors are supported by NSF CCF-1740850, CCF-1839317, CCF-2402572, and DMS-2023495.

Cite As Get BibTex

Daniel Paul-Pena and C. Seshadhri. A Dichotomy Theorem for Linear Time Homomorphism Orbit Counting in Bounded Degeneracy Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.54

Abstract

Counting the number of homomorphisms of a pattern graph H in a large input graph G is a fundamental problem in computer science. In many applications in databases, bioinformatics, and network science, we need more than just the total count. We wish to compute, for each vertex v of G, the number of H-homomorphisms that v participates in. This problem is referred to as homomorphism orbit counting, as it relates to the orbits of vertices of H under its automorphisms.
Given the need for fast algorithms for this problem, we study when near-linear time algorithms are possible. A natural restriction is to assume that the input graph G has bounded degeneracy, a commonly observed property in modern massive networks. Can we characterize the patterns H for which homomorphism orbit counting can be done in near-linear time?
We discover a dichotomy theorem that resolves this problem. For pattern H, let 𝓁 be the length of the longest induced path between any two vertices of the same orbit (under the automorphisms of H). If 𝓁 ≤ 5, then H-homomorphism orbit counting can be done in near-linear time for bounded degeneracy graphs. If 𝓁 > 5, then (assuming fine-grained complexity conjectures) there is no near-linear time algorithm for this problem. We build on existing work on dichotomy theorems for counting the total H-homomorphism count. Surprisingly, there exist (and we characterize) patterns H for which the total homomorphism count can be computed in near-linear time, but the corresponding orbit counting problem cannot be done in near-linear time.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Homomorphism counting
  • Bounded degeneracy graphs
  • Fine-grained complexity
  • Orbit counting
  • Subgraph counting

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proc. 55th Annual IEEE Symposium on Foundations of Computer Science, 2014. URL: https://doi.org/10.1109/FOCS.2014.53.
  2. Nesreen K. Ahmed, Jennifer Neville, Ryan A. Rossi, and Nick Duffield. Efficient graphlet counting for large networks. In Proceedings, SIAM International Conference on Data Mining (ICDM), 2015. URL: https://doi.org/10.1109/ICDM.2015.141.
  3. Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. URL: https://doi.org/10.1007/BF02523189.
  4. A. Benson, D. F. Gleich, and J. Leskovec. Higher-order organization of complex networks. Science, 353(6295):163-166, 2016. URL: https://doi.org/10.1126/science.aad9029.
  5. Suman K Bera, Amit Chakrabarti, and Prantar Ghosh. Graph coloring via degeneracy in streaming and other space-conscious models. In International Colloquium on Automata, Languages and Programming, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.11.
  6. Suman K. Bera, Lior Gishboliner, Yevgeny Levanzov, C. Seshadhri, and Asaf Shapira. Counting subgraphs in degenerate graphs. Journal of the ACM (JACM), 69(3), 2022. URL: https://doi.org/10.1145/3520240.
  7. Suman K Bera, Noujan Pashanasangi, and C Seshadhri. Linear time subgraph counting, graph degeneracy, and the chasm at size six. In Proc. 11th Conference on Innovations in Theoretical Computer Science. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.38.
  8. Suman K. Bera, Noujan Pashanasangi, and C. Seshadhri. Near-linear time homomorphism counting in bounded degeneracy graphs: The barrier of long induced cycles. In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2315-2332, 2021. URL: https://doi.org/10.1137/1.9781611976465.138.
  9. Suman K Bera and C Seshadhri. How the degeneracy helps for triangle counting in graph streams. In Principles of Database Systems, pages 457-467, 2020. URL: https://doi.org/10.1145/3375395.3387665.
  10. Jonathan W. Berry, Bruce Hendrickson, Randall A. LaViolette, and Cynthia A. Phillips. Tolerating the community detection resolution limit with edge weighting. Phys. Rev. E, 83:056119, 2011. URL: https://doi.org/10.1103/PhysRevE.83.056119.
  11. Umberto Bertele and Francesco Brioschi. On non-serial dynamic programming. J. Comb. Theory, Ser. A, 14(2):137-148, 1973. URL: https://doi.org/10.1016/0097-3165(73)90016-2.
  12. J.A. Bondy and U.S.R Murty. Graph Theory, volume 244. Springer, 2008. URL: https://doi.org/10.1007/978-1-84628-970-5.
  13. Christian Borgs, Jennifer Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Counting graph homomorphisms. In Topics in discrete mathematics, pages 315-371. Springer, 2006. URL: https://doi.org/10.1007/3-540-33700-8_18.
  14. Marco Bressan. Faster subgraph counting in sparse graphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.IPEC.2019.6.
  15. Marco Bressan. Faster algorithms for counting subgraphs in sparse graphs. Algorithmica, 83:2578-2605, 2021. URL: https://doi.org/10.1007/s00453-021-00811-0.
  16. Marco Bressan, Leslie Ann Goldberg, Kitty Meeks, and Marc Roth. Counting subgraphs in somewhere dense graphs. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), pages 27:1-27:14, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.27.
  17. Marco Bressan and Marc Roth. Exact and approximate pattern counting in degenerate graphs: New algorithms, hardness results, and complexity dichotomies. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 276-285, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00036.
  18. Graham R Brightwell and Peter Winkler. Graph homomorphisms and phase transitions. Journal of combinatorial theory, series B, 77(2):221-262, 1999. URL: https://doi.org/10.1006/jctb.1999.1899.
  19. Ashok K Chandra and Philip M Merlin. Optimal implementation of conjunctive queries in relational data bases. In Proc. 9th Annual ACM Symposium on the Theory of Computing, pages 77-90, 1977. URL: https://doi.org/10.1145/800105.803397.
  20. Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing (SICOMP), 14(1):210-223, 1985. URL: https://doi.org/10.1137/0214017.
  21. Jonathan Cohen. Graph twiddling in a mapreduce world. Computing in Science & Engineering, 11(4):29, 2009. URL: https://doi.org/10.1109/MCSE.2009.120.
  22. J. Coleman. Social capital in the creation of human capital. American Journal of Sociology, 94:S95-S120, 1988. URL: https://doi.org/10.1086/228943.
  23. 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, pages 210-223, 2017. URL: https://doi.org/10.1145/3055399.3055502.
  24. Víctor Dalmau and Peter Jonsson. The complexity of counting homomorphisms seen from the other side. Theoretical Computer Science, 329(1-3):315-323, 2004. URL: https://doi.org/10.1016/j.tcs.2004.08.008.
  25. Holger Dell, Marc Roth, and Philip Wellnitz. Counting answers to existential questions. In Proc. 46th International Colloquium on Automata, Languages and Programming. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.113.
  26. Josep Díaz, Maria Serna, and Dimitrios M Thilikos. Counting h-colorings of partial k-trees. Theoretical Computer Science, 281(1-2):291-309, 2002. URL: https://doi.org/10.1016/S0304-3975(02)00017-8.
  27. Martin Dyer and Catherine Greenhill. The complexity of counting graph homomorphisms. Random Structures & Algorithms, 17(3-4):260-289, 2000. URL: https://doi.org/10.1002/1098-2418(200010/12)17:3/4%3C260::AID-RSA5%3E3.0.CO;2-W.
  28. David Eppstein. Arboricity and bipartite subgraph listing algorithms. Information processing letters, 51(4):207-211, 1994. URL: https://doi.org/10.1016/0020-0190(94)90121-X.
  29. G. Fagiolo. Clustering in complex directed networks. Phys. Rev. E, 2007. URL: https://doi.org/10.1103/PhysRevE.76.026107.
  30. Jörg Flum and Martin Grohe. The parameterized complexity of counting problems. SIAM Journal on Computing (SICOMP), 33(4):892-922, 2004. URL: https://doi.org/10.1137/S0097539703427203.
  31. Lior Gishboliner, Yevgeny Levanzov, and Asaf Shapira. Counting subgraphs in degenerate graphs, 2020. https://arxiv.org/abs/2010.05998, URL: https://doi.org/10.48550/arXiv.2010.05998.
  32. Gaurav Goel and Jens Gustedt. Bounded arboricity to determine the local structure of sparse graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 159-167. Springer, 2006. URL: https://doi.org/10.1007/11917496_15.
  33. Rudolf Halin. S-functions for graphs. Journal of geometry, 8(1-2):171-186, 1976. URL: https://doi.org/10.1007/BF01917434.
  34. P. Holland and S. Leinhardt. A method for detecting structure in sociometric data. American Journal of Sociology, 76:492-513, 1970. URL: https://doi.org/10.1016/B978-0-12-442450-0.50028-6.
  35. Alon Itai and Michael Rodeh. Finding a minimum circuit in a graph. SIAM Journal on Computing, 7(4):413-423, 1978. URL: https://doi.org/10.1137/0207033.
  36. Shalev Itzkovitz, Reuven Levitt, Nadav Kashtan, Ron Milo, Michael Itzkovitz, and Uri Alon. Coarse-graining and self-dissimilarity of complex networks. Phys. Rev. E, 71(016127), January 2005. URL: https://doi.org/10.1103/PhysRevE.71.016127.
  37. Shweta Jain and C Seshadhri. A fast and provable method for estimating clique counts using Turán’s theorem. In Proceedings, International World Wide Web Conference (WWW), pages 441-449, 2017. URL: https://doi.org/10.1145/3038912.3052636.
  38. Madhav Jha, C Seshadhri, and Ali Pinar. Path sampling: A fast and provable method for estimating 4-vertex subgraph counts. In Proc. 24th Proceedings, International World Wide Web Conference (WWW), pages 495-505. International World Wide Web Conferences Steering Committee, 2015. URL: https://doi.org/10.1145/2736277.2741101.
  39. László Lovász. Operations with structures. Acta Mathematica Academiae Scientiarum Hungarica, 18(3-4):321-328, 1967. URL: https://doi.org/10.1007/BF02280291.
  40. László Lovász. Large networks and graph limits, volume 60. American Mathematical Soc., 2012. URL: http://www.ams.org/bookstore-getitem/item=COLL-60.
  41. David W Matula and Leland L Beck. Smallest-last ordering and clustering and graph coloring algorithms. Journal of the ACM (JACM), 30(3):417-427, 1983. URL: https://doi.org/10.1145/2402.322385.
  42. Derek O'Callaghan, Martin Harrigan, Joe Carthy, and Pádraig Cunningham. Identifying discriminating network motifs in youtube spam, 2012. https://arxiv.org/abs/1202.5216, URL: https://doi.org/10.48550/arXiv.1202.5216.
  43. Mark Ortmann and Ulrik Brandes. Efficient orbit-aware triad and quad census in directed and undirected graphs. Applied network science, 2(1), 2017. URL: https://doi.org/10.1007/s41109-017-0027-2.
  44. Noujan Pashanasangi and C Seshadhri. Efficiently counting vertex orbits of all 5-vertex subgraphs, by evoke. In Proc. 13th International Conference on Web Search and Data Mining (WSDM), pages 447-455, 2020. URL: https://doi.org/10.1145/3336191.3371773.
  45. Ali Pinar, C Seshadhri, and Vaidyanathan Vishal. Escape: Efficiently counting all 5-vertex subgraphs. In Proceedings, International World Wide Web Conference (WWW), pages 1431-1440, 2017. URL: https://doi.org/10.1145/3038912.3052597.
  46. Natasa Przulj. Biological network comparison using graphlet degree distribution. Bioinformatics, 23(2):177-183, 2007. URL: https://doi.org/10.1093/bioinformatics/btl301.
  47. Neil Robertson and Paul D. Seymour. Graph minors. i. excluding a forest. Journal of Combinatorial Theory, Series B, 35(1):39-61, 1983. URL: https://doi.org/10.1016/0095-8956(83)90079-5.
  48. Neil Robertson and Paul D. Seymour. Graph minors. iii. planar tree-width. Journal of Combinatorial Theory, Series B, 36(1):49-64, 1984. URL: https://doi.org/10.1016/0095-8956(84)90013-3.
  49. Neil Robertson and Paul D. Seymour. Graph minors. ii. algorithmic aspects of tree-width. Journal of algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  50. Rahmtin Rotabi, Krishna Kamath, Jon M. Kleinberg, and Aneesh Sharma. Detecting strong ties using network motifs. In Proceedings, International World Wide Web Conference (WWW), 2017. URL: https://doi.org/10.1145/3041021.3055139.
  51. Marc Roth and Philip Wellnitz. Counting and finding homomorphisms is universal for parameterized complexity theory. In Proc. 31st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2161-2180, 2020. URL: https://doi.org/10.1137/1.9781611975994.133.
  52. Ahmet Erdem Sariyuce, C. Seshadhri, Ali Pinar, and Umit V. Catalyurek. Finding the hierarchy of dense subgraphs using nucleus decompositions. In Proceedings, International World Wide Web Conference (WWW), pages 927-937, 2015. URL: https://doi.org/10.1145/2736277.2741640.
  53. C. Seshadhri and Srikanta Tirthapura. Scalable subgraph counting: The methods behind the madness: WWW 2019 tutorial. In Proceedings, International World Wide Web Conference (WWW), 2019. URL: https://doi.org/10.1145/3308560.3320092.
  54. Nino Shervashidze, S. V. N. Vishwanathan, Tobias Petri, Kurt Mehlhorn, and Karsten M. Borgwardt. Efficient graphlet kernels for large graph comparison. In AISTATS, pages 488-495, 2009. URL: http://proceedings.mlr.press/v5/shervashidze09a.html.
  55. K. Shin, T. Eliassi-Rad, and C. Faloutsos. Patterns and anomalies in k-cores of real-world graphs with applications. Knowledge and Information Systems, 54(3):677-710, 2018. URL: https://doi.org/10.1007/s10115-017-1077-6.
  56. George Szekeres and Herbert S Wilf. An inequality for the chromatic number of a graph. Journal of Combinatorial Theory, 4(1):1-3, 1968. URL: https://doi.org/10.1016/S0021-9800(68)80081-X.
  57. Charalampos E. Tsourakakis. The k-clique densest subgraph problem. In Proceedings, International World Wide Web Conference (WWW), pages 1122-1132, 2015. URL: https://doi.org/10.1145/2736277.2741098.
  58. Charalampos E. Tsourakakis, Jakub Pachocki, and Michael Mitzenmacher. Scalable motif-aware graph clustering. In Proceedings, International World Wide Web Conference (WWW), pages 1451-1460, 2017. URL: https://doi.org/10.1145/3038912.3052653.
  59. Johan Ugander, Lars Backstrom, and Jon M. Kleinberg. Subgraph frequencies: mapping the empirical and extremal geography of large graph collections. In Proceedings, International World Wide Web Conference (WWW), pages 1307-1318, 2013. URL: https://doi.org/10.1145/2488388.2488502.
  60. Hao Yin, Austin R. Benson, and Jure Leskovec. Higher-order clustering in networks. Phys. Rev. E, 97:052306, 2018. URL: https://doi.org/10.1103/PhysRevE.97.052306.
  61. Hao Yin, Austin R. Benson, and Jure Leskovec. The local closure coefficient: A new perspective on network clustering. In ACM International Conference on Web Search and Data Mining (WSDM), pages 303-311, 2019. URL: https://doi.org/10.1145/3289600.3290991.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact