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.
Similar content being viewed by others
References
N. Alon: The Shannon capacity of a union, Combinatorica18 (1998), 301–310.
B. Bukh and C. Cox: On a fractional version of Haemers’ bound, arXiv:1802.00476, 2018.
P. Bürgisser, M. Clausen and M. Amin Shokrollahi: Algebraic complexity theory, volume 315 of Grundlehren Math. Wiss. Springer-Verlag, Berlin, 1997.
A. Blasiak: A graph-theoretic approach to network coding, PhD thesis, Cornell University, 2013.
E. Becker and N. Schwartz: Zum Darstellungssatz von Kadison-Dubois, Arch. Math. (Basel)40 (1983), 421–428.
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.
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.
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.
T. Fritz: Resource convertibility and ordered commutative monoids, Math. Structures Comput. Sci.27 (2017), 850–938.
W. Haemers: On some problems of Lovasz concerning the Shannon capacity of a graph, IEEE Trans. Inform. Theory25 (1979), 231–232.
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.
D. E. Knuth: The sandwich theorem, Electron. J. Combin., 1(1): 1, 1994.
L. Lovász: On the Shannon capacity of a graph, IEEE Trans. Inform. Theory25 (1979), 1–7.
M. Marshall: Positive polynomials and sums of squares, volume 146 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2008.
R. J. McEliece and E. C. Posner: Hide and seek, data storage, and entropy, The Annals of Mathematical Statistics42 (1971), 1706–1716.
A. Prestel and Ch. N. Delzell: Positive polynomials, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2001, From Hilbert’s 17th problem to real algebra.
A. Schrijver: Combinatorial optimization: polyhedra and efficiency, volume 24, Springer Science & Business Media, 2003.
C. E. Shannon: The zero error capacity of a noisy channel, Institute of Radio Engineers, Transactions on Information TheoryIT-2 (1956), 8–19.
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.
V. Strassen: Relative bilinear complexity and matrix multiplication, J. Reine Angew. Math.375/376 (1987), 406–443.
V. Strassen: The asymptotic spectrum of tensors, J. Peine Angew. Math.384 (1988), 102–152.
V. Strassen: Degeneration and complexity of bilinear maps: some asymptotic spectra, J. Peine Angew. Math.413 (1991), 127–180.
T. Tao: A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound, https://terrytao.wordpress.com, 2016.
L. Wang and O. Shayevitz: Graph information ratio, SIAM Journal on Discrete Mathematics31 (2017), 2703–2734.
J. Zuiddam: Asymptotic spectra, algebraic complexity and moment polytopes, PhD thesis, University of Amsterdam, 2018.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-019-3992-5