Skip to main content
SpringerLink
Log in

Approximation of the Shannon Capacity Via Matrix Cone Programming

  • Published:
Journal of the Operations Research Society of China Aims and scope Submit manuscript

Abstract

This paper proposes a novel formulation using the matrix cone programming to compute an upper bound of the Shannon capacity of graphs, which is theoretically superior to the Lovász number. To achieve this, a sequence of matrix cones is constructed by adding certain co-positive matrices to the positive semi-definite matrix cones during the matrix cone programming. We require the sequence of matrix cones to have the weak product property so that the improved result of the matrix cone programming remains an upper bound of the Shannon capacity. Our result shows that the existence of a sequence of suitable matrix cones with the weak product property is equivalent to the existence of a co-positive matrix with testable conditions. Finally, we give some concrete examples with special structures to verify the existence of the matrix cone sequence.

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. Shannon, C.E.: The zero error capacity of a noisy channel. IRE Trans. Inform. Theory 2(3), 8–19 (1956)

    Article  MathSciNet  Google Scholar 

  2. Alon, N., Lubetzky, E.: The Shannon capacity of a graph and the independence numbers of its powers. IEEE Trans. Inf. Theory 52(5), 2172–2176 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lovász, László.: Graphs and Geometry. American Society of Mathematics (2019) https://doi.org/10.1090/coll/065

  5. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim.n Methods Softw. 11(1), 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3 - a MATLAB software package for semidefinite programming, version 1.3. Optim. Methods Softw. 11(1), 545–581 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Sun, D., Toh, K.C., Yuan, Y., Zhao, X.Y.: SDPNAL+: a Matlab software for semidefinite programming with bound constraints (version 1.0). Optim. Methods Softw. 35(1), 87–115 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Schrijver, A.: A Comparison of the Delsarte and Lovász Bounds. IEEE Trans. Inf. Theory 25(4), 425–429 (1979)

    Article  MATH  Google Scholar 

  9. De Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Parrilo, P. A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and Optimization. Technical report, Massachusetts Institute of Technology (2000)

  11. Peña, J., Vera, J., Zuluaga, L.F.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18(1), 87–105 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bomze, I.M., Frommlet, F., Locatelli, M.: Copositivity cuts for improving SDP bounds on the clique number. Math. Program. 124(1–2), 13–32 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bi, Y., Tang, A.: On upper bounding Shannon capacity of graph through generalized conic programming. Optim. Lett. 13(6), 1313–1323 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Polak, S.C., Schrijver, A.: New lower bound on the Shannon capacity of \(C_7\) from circular graphs. Inf. Process. Lett. 143(2), 37–40 (2019)

    Article  MATH  Google Scholar 

  15. Zhao, X.Y., Sun, D., Toh, K.C.: A newton-cg augmented lagrangian method for semidefinite programming. SIAM J. Optim. 20(4), 1737–1765 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, L., Sun, D., Toh, K.C.: SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints. Math. Program. Comput. 7(3), 331–366 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Haemers, W.: On Some Problems of Lovasz Concerning the Shannon Capacity of a Graph. IEEE Trans. Inf. Theory 25(2), 231–232 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Haemers, W.H.: An upper bound for the shannon capacity of a graph. Colloquia Math. Societatis Janos Bolyai 25, 267–272 (1978)

    MathSciNet  Google Scholar 

  19. Sihuang, H., Tamo, I., Shayevitz, O.: A bound on the shannon capacity via a linear programming variation. SIAM J. Discret. Math. 32(3), 2229–2241 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Prof. Si-Huang Hu, Prof. De-Feng Sun and Prof. Hao Wu for their valuable discussions and helpful advice on optimization and information theory.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Shi-Tong Wu & Zhong-Yi Huang

  2. Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co., Ltd., Hong Kong, China

    Shi-Tong Wu & Bo Bai

  3. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

    Zhen-Nan Zhou

Authors
  1. Shi-Tong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen-Nan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhong-Yi Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bo Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Shi-Tong Wu and Zhen-Nan Zhou developed the presented idea of the matrix cone programming to investigate the Shannon Capacity. Shi-Tong Wu drafted the article, Zhen-Nan Zhou, Zhong-Yi Huang and Bo Bai reviewed and co-supervised this work. Zhen-Nan Zhou and Zhong-Yi Huang verified the theoretical result and examples and supervised the findings of this work. Bo Bai provided useful suggestions on the information theory and the application of the Shannon Capacity from industrial perspective. All the authors discussed the results and contributed to the final manuscript.

Corresponding author

Correspondence to Zhen-Nan Zhou.

Ethics declarations

Conflict of intersts

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11871297, 11871298, 12025104 and 12031013) and the Tsinghua University Initiative Scientific Research Program. Zhen-Nan Zhou was also partially supported by the National Key R &D Program of China (No. 2020YFA0712000).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, ST., Zhou, ZN., Huang, ZY. et al. Approximation of the Shannon Capacity Via Matrix Cone Programming. J. Oper. Res. Soc. China 11, 875–889 (2023). https://doi.org/10.1007/s40305-022-00408-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40305-022-00408-6

Keywords

Mathematics Subject Classification

Search

Navigation