Approximation of the Shannon Capacity Via Matrix Cone Programming

doi:10.1007/s40305-022-00408-6

Journal of the Operations Research Society of China ›› 2023, Vol. 11 ›› Issue (4): 875-889.doi: 10.1007/s40305-022-00408-6

收稿日期:2021-02-04 修回日期:2022-02-18 出版日期:2023-12-30 发布日期:2023-12-26
通讯作者: Zhen-Nan Zhou, Shi-Tong Wu, Zhong-Yi Huang, Bo Bai E-mail:zhennan@bicmr.pku.edu.cn;wust20@mails.tsinghua.edu.cn;zhongyih@tsinghua.edu.cn;ee.bobbai@gmail.com

Approximation of the Shannon Capacity Via Matrix Cone Programming

Shi-Tong Wu^1,2, Zhen-Nan Zhou³, Zhong-Yi Huang¹, Bo Bai²

1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China;
2. Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co., Ltd., Hong Kong, China;
3. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

Received:2021-02-04 Revised:2022-02-18 Online:2023-12-30 Published:2023-12-26
Contact: Zhen-Nan Zhou, Shi-Tong Wu, Zhong-Yi Huang, Bo Bai E-mail:zhennan@bicmr.pku.edu.cn;wust20@mails.tsinghua.edu.cn;zhongyih@tsinghua.edu.cn;ee.bobbai@gmail.com
Supported by:
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).

摘要/Abstract

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.

Key words: Shannon capacity, Lovász number, Matrix cone programming, Weak product property

中图分类号:

. [J]. Journal of the Operations Research Society of China, 2023, 11(4): 875-889.

Shi-Tong Wu, Zhen-Nan Zhou, Zhong-Yi Huang, Bo Bai. Approximation of the Shannon Capacity Via Matrix Cone Programming[J]. Journal of the Operations Research Society of China, 2023, 11(4): 875-889.

参考文献

[1] Shannon, C.E.: The zero error capacity of a noisy channel. IRE Trans. Inform. Theory 2(3), 8-19 (1956)
[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)
[3] Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1-7 (1979)
[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)
[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)
[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)
[8] Schrijver, A.: A Comparison of the Delsarte and Lovász Bounds. IEEE Trans. Inf. Theory 25(4), 425-429 (1979)
[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)
[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)
[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)
[13] Bi, Y., Tang, A.: On upper bounding Shannon capacity of graph through generalized conic programming. Optim. Lett. 13(6), 1313-1323 (2019)
[14] Polak, S.C., Schrijver, A.: New lower bound on the Shannon capacity of C₇ from circular graphs. Inf. Process. Lett. 143(2), 37-40 (2019)
[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)
[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)
[17] Haemers, W.: On Some Problems of Lovasz Concerning the Shannon Capacity of a Graph. IEEE Trans. Inf. Theory 25(2), 231-232 (1979)
[18] Haemers, W.H.: An upper bound for the shannon capacity of a graph. Colloquia Math. Societatis Janos Bolyai 25, 267-272 (1978)
[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)

Approximation of the Shannon Capacity Via Matrix Cone Programming

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 11

编辑推荐

Metrics

本文评价