Document Open Access Logo

On the Expressive Power of Linear Algebra on Graphs

Author Floris Geerts

PDF
Thumbnail PDF

File

LIPIcs.ICDT.2019.7.pdf
  • Filesize: 0.6 MB
  • 19 pages

Document Identifiers

Author Details

Floris Geerts
  • University of Antwerp, Antwerp, Belgium

Cite AsGet BibTex

Floris Geerts. On the Expressive Power of Linear Algebra on Graphs. In 22nd International Conference on Database Theory (ICDT 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 127, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICDT.2019.7

Abstract

Most graph query languages are rooted in logic. By contrast, in this paper we consider graph query languages rooted in linear algebra. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported. We investigate the problem of characterising equivalence of graphs, represented by their adjacency matrices, for various fragments of MATLANG. A complete picture is painted of the impact of the linear algebra operations in MATLANG on their ability to distinguish graphs.

Subject Classification

ACM Subject Classification
  • Information systems → Query languages
  • Mathematics of computing → Graph theory
  • Theory of computation → Logic
Keywords
  • matrix query languages
  • graph queries
  • graph theory
  • logics

Metrics

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

Related Versions

References

  1. Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. URL: http://dx.doi.org/10.1007/BF02523189.
  2. Renzo Angles, Marcelo Arenas, Pablo Barceló, Aidan Hogan, Juan Reutter, and Domagoj Vrgoč. Foundations of Modern Query Languages for Graph Databases. ACM Comput. Surv., 50(5):68:1-68:40, 2017. URL: http://dx.doi.org/10.1145/3104031.
  3. Albert Atserias and Elitza N. Maneva. Sherali-Adams Relaxations and Indistinguishability in Counting Logics. SIAM J. Comput., 42(1):112-137, 2013. URL: http://dx.doi.org/10.1137/120867834.
  4. Sheldon Axler. Linear Algebra Done Right. Springer, third edition, 2015. URL: http://dx.doi.org/10.1007/978-3-319-11080-6.
  5. Pablo Barceló. Querying Graph Databases. In Proceedings of the 32nd ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS, pages 175-188, 2013. URL: http://dx.doi.org/10.1145/2463664.2465216.
  6. Oliver Bastert. Stabilization procedures and applications. PhD thesis, Technical University Munich, Germany, 2001. URL: http://nbn-resolving.de/urn:nbn:de:bvb:91-diss2002070500045.
  7. Matthias Boehm, Michael W. Dusenberry, Deron Eriksson, Alexandre V. Evfimievski, Faraz Makari Manshadi, Niketan Pansare, Berthold Reinwald, Frederick R. Reiss, Prithviraj Sen, Arvind C. Surve, and Shirsih Tatikonda. SystemML: Declarative machine learning on Spark. Proceedings of the VLDB Endowment, 9(13):1425-1436, 2016. URL: http://dx.doi.org/10.14778/3007263.3007279.
  8. Matthias Boehm, Berthold Reinwald, Dylan Hutchison, Prithviraj Sen, Alexandre V. Evfimievski, and Niketan Pansare. On Optimizing Operator Fusion Plans for Large-scale Machine Learning in SystemML. Proceedings of the VLDB Endowment, 11(12):1755-1768, 2018. URL: http://dx.doi.org/10.14778/3229863.3229865.
  9. Robert Brijder, Floris Geerts, Jan Van den Bussche, and Timmy Weerwag. On the Expressive Power of Query Languages for Matrices. In 21st International Conference on Database Theory, ICDT, pages 10:1-10:17, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICDT.2018.10.
  10. Andries E. Brouwer and Willem H. Haemers. Spectra of Graphs. Universitext. Springer, 2012. URL: http://dx.doi.org/10.1007/978-1-4614-1939-6.
  11. Jin-Yi Cai, Martin Fürer, and Neil Immerman. An optimal lower bound on the number of variables for graph identification. Combinatorica, 12(4):389-410, 1992. URL: http://dx.doi.org/10.1007/BF01305232.
  12. Lingjiao Chen, Arun Kumar, Jeffrey Naughton, and Jignesh M. Patel. Towards Linear Algebra over Normalized Data. Proceedings of the VLDB Endowment, 10(11):1214-1225, 2017. URL: http://dx.doi.org/10.14778/3137628.3137633.
  13. Dragoš M. Cvetković. GRAPHS AND THEIR SPECTRA. Publikacije Elektrotehničkog fakulteta. Serija Matematika i fizika, 354/356:1-50, 1971. URL: http://www.jstor.org/stable/43667526.
  14. Dragoš M. Cvetković. The main part of the spectrum, divisors and switching of graphs. Publ. Inst. Math. (Beograd) (N.S.), 23(37):31-38, 1978. URL: http://elib.mi.sanu.ac.rs/files/journals/publ/43/6.pdf.
  15. Dragoš M. Cvetković, Peter Rowlinson, and Slobodan Simić. Eigenspaces of Graphs. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1997. URL: http://dx.doi.org/10.1017/CBO9781139086547.
  16. Dragoš M. Cvetković, Peter Rowlinson, and Slobodan Simić. An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts. Cambridge University Press, 2009. URL: http://dx.doi.org/10.1017/CBO9780511801518.
  17. Anuj Dawar. On the Descriptive Complexity of Linear Algebra. In Proceedings of the 15th International Workshop on Logic, Language, Information and Computation, WoLLIC, pages 17-25, 2008. URL: http://dx.doi.org/10.1007/978-3-540-69937-8_2.
  18. Anuj Dawar, Martin Grohe, Bjarki Holm, and Bastian Laubner. Logics with Rank Operators. In Proceedings of the 24th Annual IEEE Symposium on Logic In Computer Science, LICS, pages 113-122, 2009. URL: http://dx.doi.org/10.1109/LICS.2009.24.
  19. Anuj Dawar and Bjarki Holm. Pebble games with algebraic rules. Fund. Inform., 150(3-4):281-316, 2017. URL: http://dx.doi.org/10.3233/FI-2017-1471.
  20. Anuj Dawar, Simone Severini, and Octavio Zapata. Descriptive Complexity of Graph Spectra. In Proceedings of the 23rd International Workshop on Logic, Language, Information and Computation, WoLLIC, pages 183-199, 2016. URL: http://dx.doi.org/10.1007/978-3-662-52921-8_12.
  21. Holger Dell, Martin Grohe, and Gaurav Rattan. Lovász Meets Weisfeiler and Lehman. In 45th International Colloquium on Automata, Languages, and Programming, ICALP, pages 40:1-40:14, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.40.
  22. Ahmed Elgohary, Matthias Boehm, Peter J. Haas, Frederick R. Reiss, and Berthold Reinwald. Compressed linear algebra for large-scale machine learning. The VLDB Journal, pages 1-26, 2017. URL: http://dx.doi.org/10.1007/s00778-017-0478-1.
  23. Shmuel Friedland. Coherent algebras and the graph isomorphism problem. Discrete Applied Mathematics, 25(1):73-98, 1989. URL: http://dx.doi.org/10.1016/0166-218X(89)90047-4.
  24. Martin Fürer. On the power of combinatorial and spectral invariants. Linear Algebra and its Applications, 432(9):2373-2380, 2010. URL: http://dx.doi.org/10.1016/j.laa.2009.07.019.
  25. Martin Fürer. On the Combinatorial Power of the Weisfeiler-Lehman Algorithm. In Dimitris Fotakis, Aris Pagourtzis, and Vangelis Th. Paschos, editors, Algorithms and Complexity, pages 260-271. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-57586-5_22.
  26. Chris Godsil and Gordon F. Royle. Algebraic Graph Theory, volume 207 of Graduate Texts in Mathematics. Springer, 2001. URL: http://dx.doi.org/10.1007/978-1-4613-0163-9.
  27. Erich Grädel and Wied Pakusa. Rank Logic is Dead, Long Live Rank Logic! In 24th EACSL Annual Conference on Computer Science Logic, CSL, pages 390-404, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CSL.2015.390.
  28. Martin Grohe, Kristian Kersting, Martin Mladenov, and Erkal Selman. Dimension Reduction via Colour Refinement. In 22th Annual European Symposium on Algorithms, ESA, pages 505-516, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_42.
  29. Martin Grohe and Martin Otto. Pebble games and linear equations. The Journal of Symbolic Logic, 80(3):797-844, 2015. URL: http://dx.doi.org/10.1017/jsl.2015.28.
  30. Martin Grohe and Wied Pakusa. Descriptive complexity of linear equation systems and applications to propositional proof complexity. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS, pages 1-12, 2017. URL: http://dx.doi.org/10.1109/LICS.2017.8005081.
  31. Willem H. Haemers and Edward Spence. Enumeration of cospectral graphs. European J. Combin., 25(2):199-211, 2004. URL: http://dx.doi.org/10.1016/S0195-6698(03)00100-8.
  32. Frank Harary and Allen J. Schwenk. The spectral approach to determining the number of walks in a graph. Pacific J. Math., 80(2):443-449, 1979. URL: https://projecteuclid.org:443/euclid.pjm/1102785717.
  33. Donald G. Higman. Coherent algebras. Linear Algebra and its Applications, 93:209-239, 1987. URL: http://dx.doi.org/10.1016/S0024-3795(87)90326-0.
  34. Bjarki Holm. Descriptive Complexity of Linear Algebra. PhD thesis, University of Cambridge, 2010. Google Scholar
  35. Dylan Hutchison, Bill Howe, and Dan Suciu. LaraDB: A Minimalist Kernel for Linear and Relational Algebra Computation. In Proceedings of the 4th ACM SIGMOD Workshop on Algorithms and Systems for MapReduce and Beyond, BeyondMR, pages 2:1-2:10, 2017. URL: http://dx.doi.org/10.1145/3070607.3070608.
  36. Neil Immerman and Eric Lander. Describing Graphs: A First-Order Approach to Graph Canonization. In Alan L. Selman, editor, Complexity Theory Retrospective: In Honor of Juris Hartmanis on the Occasion of His Sixtieth Birthday, pages 59-81. Springer, 1990. URL: http://dx.doi.org/10.1007/978-1-4612-4478-3_5.
  37. Naihuan Jing. Unitary and orthogonal equivalence of sets of matrices. Linear Algebra and its Applications, 481:235-242, 2015. URL: http://dx.doi.org/10.1016/j.laa.2015.04.036.
  38. Charles R Johnson and Morris Newman. A note on cospectral graphs. Journal of Combinatorial Theory, Series B, 28(1):96-103, 1980. URL: http://dx.doi.org/10.1016/0095-8956(80)90058-1.
  39. Kristian Kersting, Martin Mladenov, Roman Garnett, and Martin Grohe. Power Iterated Color Refinement. In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, AAAI'14, pages 1904-1910. AAAI Press, 2014. URL: https://www.aaai.org/ocs/index.php/AAAI/AAAI14/paper/view/8377.
  40. Andreas Kunft, Alexander Alexandrov, Asterios Katsifodimos, and Volker Markl. Bridging the Gap: Towards Optimization Across Linear and Relational Algebra. In Proceedings of the 3rd ACM SIGMOD Workshop on Algorithms and Systems for MapReduce and Beyond, BeyondMR, pages 1:1-1:4, 2016. URL: http://dx.doi.org/10.1145/2926534.2926540.
  41. Andreas Kunft, Asterios Katsifodimos, Sebastian Schelter, Tilmann Rabl, and Volker Markl. BlockJoin: Efficient Matrix Partitioning Through Joins. Proceedings of the VLDB Endowment, 10(13):2061-2072, 2017. URL: http://dx.doi.org/10.14778/3151106.3151110.
  42. Leonid Libkin. Elements of Finite Model Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2004. URL: http://dx.doi.org/10.1007/978-3-662-07003-1.
  43. Shangyu Luo, Zekai J. Gao, Michael Gubanov, Luis L. Perez, and Christopher Jermaine. Scalable Linear Algebra on a Relational Database System. SIGMOD Rec., 47(1):24-31, 2018. URL: http://dx.doi.org/10.1145/3277006.3277013.
  44. Peter N. Malkin. Sherali-Adams relaxations of graph isomorphism polytopes. Discrete Optimization, 12:73-97, 2014. URL: http://dx.doi.org/10.1016/j.disopt.2014.01.004.
  45. Albert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: Theory of Majorization and its Applications. Springer, 2011. URL: http://dx.doi.org/10.1007/978-0-387-68276-1.
  46. Hung Q. Ngo, XuanLong Nguyen, Dan Olteanu, and Maximilian Schleich. In-Database Factorized Learning. In Proceedings of the 11th Alberto Mendelzon International Workshop on Foundations of Data Management and the Web, AMW, 2017. URL: http://ceur-ws.org/Vol-1912/paper21.pdf.
  47. Martin Otto. Bounded Variable Logics and Counting: A Study in Finite Models, volume 9 of Lecture Notes in Logic. Cambridge University Press, 2017. URL: http://dx.doi.org/10.1017/9781316716878.
  48. Motakuri V. Ramana, Edward R. Scheinerman, and Daniel Ullman. Fractional isomorphism of graphs. Discrete Mathematics, 132(1-3):247-265, 1994. URL: http://dx.doi.org/10.1016/0012-365X(94)90241-0.
  49. Peter Rowlinson. The main eigenvalues of a graph: A survey. Applicable Analysis and Discrete Mathematics, 1(2):455-471, 2007. URL: http://www.jstor.org/stable/43666075.
  50. Edward R. Scheinerman and Daniel H. Ullman. Fractional Graph Theory: a Rational Approach to the Theory of Graphs. John Wiley &Sons, 1997. URL: https://www.ams.jhu.edu/ers/wp-content/uploads/sites/2/2015/12/fgt.pdf.
  51. Maximilian Schleich, Dan Olteanu, and Radu Ciucanu. Learning Linear Regression Models over Factorized Joins. In Proceedings of the 2016 International Conference on Management of Data, SIGMOD, pages 3-18, 2016. URL: http://dx.doi.org/10.1145/2882903.2882939.
  52. Mario Thüne. Eigenvalues of matrices and graphs. PhD thesis, University of Leipzig, 2012. Google Scholar
  53. Gottfried Tinhofer. Graph isomorphism and theorems of Birkhoff type. Computing, 36(4):285-300, 1986. URL: http://dx.doi.org/10.1007/BF02240204.
  54. Gottfried Tinhofer. A note on compact graphs. Discrete Applied Mathematics, 30(2):253-264, 1991. URL: http://dx.doi.org/10.1016/0166-218X(91)90049-3.
  55. Edwin R. van Dam and Willem H. Haemers. Which graphs are determined by their spectrum? Linear Algebra and its Applications, 373:241-272, 2003. URL: http://dx.doi.org/10.1016/S0024-3795(03)00483-X.
  56. Erwin R. van Dam, Willem H. Haemers, and Jack H. Koolen. Cospectral graphs and the generalized adjacency matrix. Linear Algebra and its Applications, 423(1):33-41, 2007. URL: http://dx.doi.org/10.1016/j.laa.2006.07.017.
  57. Boris Weisfeiler. On Construction and Identification of Graphs. Number 558 in Lecture Notes in Mathematics. Springer-Verlag Berlin Heidelberg, 1976. URL: http://dx.doi.org/10.1007/BFb0089374.
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 Imprint Privacy Contact