Skip to main content
SpringerLink
Log in

Dimension-free bounds and structural results in communication complexity

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to perhaps the more common framework in communication complexity where poly-logarithmic dependencies on the number of input bits are tolerated.

Dimension-free bounds are also closely related to structural results, where one seeks to describe the structure of Boolean matrices and functions that have low complexity. We prove such theorems for several communication and query complexity measures as well as various matrix and operator norms. In several other cases we show that such bounds do not exist.

We propose several conjectures, and establish that, in addition to applications in complexity theory, these problems are central to characterization of the idempotents of the algebra of Schur multipliers, and could lead to new extensions of Cohen’s celebrated idempotent theorem regarding the Fourier algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Alon, Covering graphs by the minimum number of equivalence relations, Combinatorica 6 (1986), 201–206.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Babai, P. Frankl and J. Simon, Complexity classes in communication complexity theory (preliminary version), in 27th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Washinton, DC, 1986, pp. 337–347.

    Google Scholar 

  3. W. D. Banks and A. Harcharras, On the norm of an idempotent Schur multiplier on the Schatten class, Proceedings of the American Mathematical Society 132 (2004), 2121–2125.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Blokhuis and T. Kloks, On the equivalence covering number of splitgraphs, Information Processing Letters 54 (1995), 301–304.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Bollobás and V. Nikiforov, Cliques and the spectral radius, Journal of Combinatorial Theory. Series B 97 (2007), 859–865.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Chudnovsky, The Erdős—Hajnal conjecture—a survey, Journal of Graph Theory 75 (2014), 178–190.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Chattopadhyay, M. Koucký, B. Loff and S. Mukhopadhyay, Simulation theorems via pseudo-random properties, Computational complexity 28 (2019), 617–659.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Chattopadhyay, S. Lovett and M. Vinyals, Equality alone does not simulate randomness, in 34th Computational Complexity Conference (CCC 2019), Leibniz International Proceedings in Informatics, Vol. 137, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Wadern, 2019, Article no. 14.

    MATH  Google Scholar 

  9. P. J. Cohen, On a conjecture of Littlewood and idempotent measures, American Journal of Mathematics 82 (1960), 191–212.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Chakrabarti and O. Regev, An optimal lower bound on the communication complexity of gap-Hamming-distance, SIAM Journal on Computing 41 (2012), 1299–1317.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. R. Davidson and A. P. Donsig, Norms of Schur multipliers, Illinois Journal of Mathematics 51 (2007), 743–766.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Duchet, Représentations, noyaux en théorie des graphes et hypergraphes, Ph.D. thesis, Université Pierre-et-Marie-Curie, Paris, 1979.

    Google Scholar 

  13. G. K. Eleftherakis, R. H. Levene and I. G. Todorov, Schur idempotents and hyperreflexivity, Israel Journal of Mathematics 215 (2016), 317–337.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bulletin de la Société Mathématique de France 92 (1964), 181–236.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Forster, A linear lower bound on the unbounded error probabilistic communication complexity, Journal of Computer and System Sciences 65 (2002), 612–625.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Frankl, Covering graphs by equivalence relations, in Theory and Practice of Combinatorics North-Holland Mathematics Studies, Vol. 60, North-Holland, Amsterdam, 1982, pp. 125–127.

    MATH  Google Scholar 

  17. M. Göös, P. Kamath, T. Pitassi and T. Watson, Query-to-communication lifting for PNP, in 32nd Computational Complexity Conference, Leibniz International Proceedings in Informatics, Vol. 79, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Wadern, 2017, Article no. 12.

    Google Scholar 

  18. M. Göös, S. Lovett, R. Meka, T. Watson and D. Zuckerman, Rectangles are nonnegative juntas, SIAM Journal on Computing 45 (2016), 1835–1869.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. D. Godsil and M. W. Newman, Eigenvalue bounds for independent sets, Journal of Combinatorial Theory. Series B 98 (2008), 721–734.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Göös, T. Pitassi and T. Watson, Query-to-communication lifting for BPP, SIAM Journal on Computing 49 (2020), 441–461.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Göös, T. Pitassi and T. Watson, Deterministic communication vs. partition number, SIAM Journal on Computing 47 (2018), 2435–2450.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Grothendieck, Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Université de Grenoble. Annales de l’Institut Fourier 4 (1952), 73–112 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  23. B. Green and T. Sanders, A quantitative version of the idempotent theorem in harmonic analysis, Annals of Mathematics 168 (2008), 1025–1054.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Gál and R. Syed, Upper bounds on communication in terms of approximate rank, in Computer Science—Theory and Applications, Lecture Notes in Computer Science, Vol. 12730, Springer, Cham, 2021, pp. 116–130.

    Chapter  Google Scholar 

  25. H. Hatami, K. Hosseini and S. Lovett, Structure of protocols for XOR functions, SIAM Journal on Computing 47 (2018), 208–217.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Host, J.-F. Méla and F. Parreau, Analyse harmonique des mesures, Astérisque 135–136 (1986).

  27. S. Jukna, On graph complexity, Combinatorics, Probability and Computing 15 (2006), 855–876.

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Kawada and K. Itô, On the probability distribution on a compact group. I, Proceedings of the Physico-Mathematical Society of Japan 22 (1940), 977–998.

    MathSciNet  MATH  Google Scholar 

  29. A. Knop, S. Lovett, S. McGuire and W. Yuan, Log-rank and lifting for AND-functions, in STOC’ 21—Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, 2021, pp. 197–208.

    Chapter  Google Scholar 

  30. E. Kushilevitz and N. Nisan, Communication Complexity, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  31. A. Katavolos and V. I. Paulsen, On the ranges of bimodule projections, Canadian Mathematical Bulletin 48 (2005), 97–111.

    Article  MathSciNet  MATH  Google Scholar 

  32. J.-L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les sphères, Advances in Mathematics 31 (1979), 16–30.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. R. Klivans and A. A. Sherstov, A lower bound for agnostically learning disjunctions, in Learning Theory, Lecture Notes in Computer Science, Vol. 4539, Springer, Berlin, 2007, pp. 409–423.

    Chapter  MATH  Google Scholar 

  34. M. Lefranc, Sur certaines algèbres de fonctions sur un groupe, C. R. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B 274 (1972), A1882–A1883.

    MathSciNet  MATH  Google Scholar 

  35. R. H. Levene, Norms of idempotent Schur multipliers, New York Journal of Mathematics 20 (2014), 325–352.

    MathSciNet  MATH  Google Scholar 

  36. L. Livshits, A note on 0–1 Schur multipliers, Linear Algebra and its Applications 222 (1995), 15–22.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Lu, H. Liu and F. Tian, Laplacian spectral bounds for clique and independence numbers of graphs, Journal of Combinatorial Theory. Series B 97 (2007), 726–732.

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Lovett, Recent advances on the log-rank conjecture in communication complexity, Bulletin of the European Association for Theoretical Computer Science 112 (2014), 18–35.

    MathSciNet  MATH  Google Scholar 

  39. T. Lee and A. Shraibman, Lower bounds in communication complexity, Foundations and Trends in Theoretical Computer Science 3 (2009), 263–399.

    Article  MathSciNet  MATH  Google Scholar 

  40. N. Linial and A. Shraibman, Lower bounds in communication complexity based on factorization norms, Random Structures & Algorithms 34 (2009), 368–394.

    Article  MathSciNet  MATH  Google Scholar 

  41. X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.

    Book  MATH  Google Scholar 

  42. J.-F. Méla, Mesures ε-idempotentes de norme bornée, Studia Mathematica 72 (1982), 131–149.

    Article  MathSciNet  MATH  Google Scholar 

  43. R. Mathias, The Hadamard operator norm of a circulant and applications, SIAM Journal on Matrix Analysis and Applications 14 (1993), 1152–1167.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Montanaro and T. Osborne, On the communication complexity of XOR functions, https://arxiv.org/abs/0909.3392.

  45. J. Mudge and H. L. Pham, Idempotents with small norms, Journal of Functional Analysis 270 (2016), 4597–4603.

    Article  MathSciNet  MATH  Google Scholar 

  46. N. S. Mande and S. Sanyal, On parity decision trees for Fourier-sparse Boolean functions, in 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, Leibniz International Proceedings in Informatics, Vol. 182, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Wadern, 2020, Article no. 29.

    Google Scholar 

  47. S. Neuwirth, Cycles and 1-unconditional matrices, Proceedings of the London Mathematical Society 93 (2006), 761–790.

    Article  MathSciNet  MATH  Google Scholar 

  48. I. Newman, Private vs. common random bits in communication complexity, Information Processing Letters 39 (1991), 67–71.

    Article  MathSciNet  MATH  Google Scholar 

  49. V. Nikiforov, The smallest eigenvalue of Kr-free graphs, Discrete Mathematics 306 (2006), 612–616.

    Article  MathSciNet  MATH  Google Scholar 

  50. V. Nikiforov, More spectral bounds on the clique and independence numbers, Journal of Combinatorial Theory. Series B 99 (2009), 819–826.

    Article  MathSciNet  MATH  Google Scholar 

  51. N. Nisan, CREW PRAMs and decision trees, SIAM Journal on Computing 20 (1991), 999–1007.

    Article  MathSciNet  MATH  Google Scholar 

  52. R. O’Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014.

    Book  MATH  Google Scholar 

  53. G. Pisier, Grothendieck’s theorem, past and present, Bulletin of the American Mathematical Society 49 (2012), 237–323.

    Article  MathSciNet  MATH  Google Scholar 

  54. P. Pudlák and V. Rüodl, Some combinatorial-algebraic problems from complexity theory, Discrete Mathematics 1 (1994), 253–279.

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Raz and P. McKenzie, Separation of the monotone NC hierarchy, in Proceedings 38th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 1997, pp. 234–243.

    Chapter  Google Scholar 

  56. S. N. Ramamoorthy and M. Sinha, On the communication complexity of greaterthan, in 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), IEEE Computer Society, Los Alamitos, CA, 2015, pp. 442–444.

    Chapter  Google Scholar 

  57. T. Sanders, A quantitative version of the non-abelian idempotent theorem, Geometric and Functional Analysis 21 (2011), 141–221.

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Sanders, Bounds in Cohen’s idempotent theorem, Journal of Fourier Analysis and Applications 26 (2020), Article no. 25.

  59. A. A. Sherstov, The pattern matrix method, SIAM Journal on Computing 40 (2011), 1969–2000.

    Article  MathSciNet  MATH  Google Scholar 

  60. A. A. Sherstov, Communication complexity theory: thirty-five years of set disjointness, in Mathematical Foundations of Computer Science 2014. Part I, Lecture Notes in Computer Science, Vol. 8634, Springer, Heidelberg, 2014, pp. 24–43.

    Chapter  Google Scholar 

  61. A. Shpilka, A. Tal and B. L. Volk, On the structure of Boolean functions with small spectral norm, Computational Complexity 26 (2017), 229–273.

    Article  MathSciNet  MATH  Google Scholar 

  62. H. Y. Tsang, C. H. Wong, N. Xie and S. Zhang, Fourier sparsity, spectral norm, and the log-rank conjecture, in Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2013, pp. 658–667.

    Chapter  Google Scholar 

  63. E. Viola, The communication complexity of addition, Combinatorica 35 (2015), 703–747.

    Article  MathSciNet  MATH  Google Scholar 

  64. A. C. C. Yao, Probabilistic computations: toward a unified measure of complexity (extended abstract), in 18th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 1977, pp. 222–227.

    Google Scholar 

  65. S. Zhang, Efficient quantum protocols for XOR-functions, in Proceedings of the Twenty-fifth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2014, pp. 1878–1885.

    Google Scholar 

  66. Z. Zhang and Y. Shi, On the parity complexity measures of Boolean functions, Theoretical Computer Science 411 (2010), 2612–2618.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, McGill University, 3480 University St., Montréal, QC, H3A 0E9, Canada

    Lianna Hambardzumyan & Hamed Hatami

  2. Department of Computer Science & Engineering, The Ohio State University, 2015 Neil Ave, Columbus, OH, 43210-1210, USA

    Pooya Hatami

Authors
  1. Lianna Hambardzumyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hamed Hatami
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pooya Hatami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pooya Hatami.

Additional information

Supported by an NSERC grant.

Supported by NSF grant CCF-1947546.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hambardzumyan, L., Hatami, H. & Hatami, P. Dimension-free bounds and structural results in communication complexity. Isr. J. Math. 253, 555–616 (2023). https://doi.org/10.1007/s11856-022-2365-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-022-2365-8

Search

Navigation