The Asymptotic Spectrum of Graphs and the Shannon Capacity

Abstract

We introduce the asymptotic spectrum of graphs and apply the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new dual characterisation of the Shannon capacity of graphs. Elements in the asymptotic spectrum of graphs include the Lovász theta number, the fractional clique cover number, the complement of the fractional orthogonal rank and the fractional Haemers bound.

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References

  1. [1]

    N. Alon: The Shannon capacity of a union, Combinatorica18 (1998), 301–310.

    MathSciNet  Article  Google Scholar 

  2. [2]

    B. Bukh and C. Cox: On a fractional version of Haemers’ bound, arXiv:1802.00476, 2018.

    Google Scholar 

  3. [3]

    P. Bürgisser, M. Clausen and M. Amin Shokrollahi: Algebraic complexity theory, volume 315 of Grundlehren Math. Wiss. Springer-Verlag, Berlin, 1997.

    Book  Google Scholar 

  4. [4]

    A. Blasiak: A graph-theoretic approach to network coding, PhD thesis, Cornell University, 2013.

    Google Scholar 

  5. [5]

    E. Becker and N. Schwartz: Zum Darstellungssatz von Kadison-Dubois, Arch. Math. (Basel)40 (1983), 421–428.

    MathSciNet  Article  Google Scholar 

  6. [6]

    T. Cubitt, L. Mančinska, D. E. Roberson, S. Severini, D. Stahlke and A. Winter: Bounds on Entanglement-Assisted Source-Channel Coding via the Lovász Theta Number and Its Variants, IEEE Trans. Inform. Theory60 (2014), 7330–7344.

    MathSciNet  Article  Google Scholar 

  7. [7]

    M. Christandl, P. Vrana and J. Zuiddam: Universal points in the asymptotic spectrum of tensors (extended abstract), in: Proceedings of 50th Annual A CM SIGACT Symposium on the Theory of Computing (STOC’18), 2018.

    Google Scholar 

  8. [8]

    J. S. Ellenberg and D. Gijswijt: On large subsets of Fq n with no three-term arithmetic progression, Ann. of Math. (2)185 (2017), 339–343.

    MathSciNet  Article  Google Scholar 

  9. [9]

    T. Fritz: Resource convertibility and ordered commutative monoids, Math. Structures Comput. Sci.27 (2017), 850–938.

    MathSciNet  Article  Google Scholar 

  10. [10]

    W. Haemers: On some problems of Lovasz concerning the Shannon capacity of a graph, IEEE Trans. Inform. Theory25 (1979), 231–232.

    MathSciNet  Article  Google Scholar 

  11. [11]

    R. M. Karp: Reducibility among combinatorial problems, in: Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N. Y., 1972), pages 85–103. Plenum, New York, 1972.

    Google Scholar 

  12. [12]

    D. E. Knuth: The sandwich theorem, Electron. J. Combin., 1(1): 1, 1994.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    L. Lovász: On the Shannon capacity of a graph, IEEE Trans. Inform. Theory25 (1979), 1–7.

    MathSciNet  Article  Google Scholar 

  14. [14]

    M. Marshall: Positive polynomials and sums of squares, volume 146 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2008.

    Google Scholar 

  15. [15]

    R. J. McEliece and E. C. Posner: Hide and seek, data storage, and entropy, The Annals of Mathematical Statistics42 (1971), 1706–1716.

    MathSciNet  Article  Google Scholar 

  16. [16]

    A. Prestel and Ch. N. Delzell: Positive polynomials, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2001, From Hilbert’s 17th problem to real algebra.

    Book  Google Scholar 

  17. [17]

    A. Schrijver: Combinatorial optimization: polyhedra and efficiency, volume 24, Springer Science & Business Media, 2003.

    MATH  Google Scholar 

  18. [18]

    C. E. Shannon: The zero error capacity of a noisy channel, Institute of Radio Engineers, Transactions on Information TheoryIT-2 (1956), 8–19.

    MathSciNet  Article  Google Scholar 

  19. [19]

    V. Strassen: The Asymptotic Spectrum of Tensors and the Exponent of Matrix Multiplication, in: Proceedings of the 27th Annual Symposium on Foundations of Computer Science, SFCS ’86, pages 49–54, Washington, DC, USA, 1986. IEEE Computer Society.

    Google Scholar 

  20. [20]

    V. Strassen: Relative bilinear complexity and matrix multiplication, J. Reine Angew. Math.375/376 (1987), 406–443.

    MathSciNet  MATH  Google Scholar 

  21. [21]

    V. Strassen: The asymptotic spectrum of tensors, J. Peine Angew. Math.384 (1988), 102–152.

    MathSciNet  MATH  Google Scholar 

  22. [22]

    V. Strassen: Degeneration and complexity of bilinear maps: some asymptotic spectra, J. Peine Angew. Math.413 (1991), 127–180.

    MathSciNet  MATH  Google Scholar 

  23. [23]

    T. Tao: A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound, https://terrytao.wordpress.com, 2016.

    Google Scholar 

  24. [24]

    L. Wang and O. Shayevitz: Graph information ratio, SIAM Journal on Discrete Mathematics31 (2017), 2703–2734.

    MathSciNet  Article  Google Scholar 

  25. [25]

    J. Zuiddam: Asymptotic spectra, algebraic complexity and moment polytopes, PhD thesis, University of Amsterdam, 2018.

    Google Scholar 

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Acknowledgements

The author thanks Harry Buhrman, Matthias Chri-standl, Péter Vrana, Jop Briët, Dion Gijswijt, Farrokh Labib, Māris Ozols, Michael Walter, Bart Sevenster, Monique Laurent, Lex Schrijver, Bart Lit-jens and the members of the A&C PhD & postdoc seminar at CWI for useful discussions and encouragement. The author is supported by NWO (617.023.116) and the QuSoft Research Center for Quantum Software. The author initiated this work when visiting the Centre for the Mathematics of Quantum Theory (QMATH) at the University of Copenhagen.

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Correspondence to Jeroen Zuiddam.

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Zuiddam, J. The Asymptotic Spectrum of Graphs and the Shannon Capacity. Combinatorica 39, 1173–1184 (2019). https://doi.org/10.1007/s00493-019-3992-5

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Mathematics Subject Classication (2010)

  • 05C69
  • 05C76
  • 94C15