# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021001
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## An optimization approach to the Langberg-Médard multiple unicast conjecture

 Department of Mathematics, University of Hong Kong, Hong Kong, no, China

Received  February 2020 Revised  November 2020 Early access March 2021

Fund Project: This research is partly supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 17301017) and a grant by the National Natural Science Foundation of China (Project No. 61871343)

The Langberg-Médard multiple unicast conjecture claims that for any strongly reachable $k$-pair network, there exists a multi-flow with rate $(1,1,\dots,1)$. In this paper, we examine an optimization problem $\mathcal P_{\mathcal S_k}$ such that its optimal value $\mathcal O_{\mathcal S_k}$ naturally gives a lower bound on the multi-flow rate for the strongly reachable $k$-pair network. We first prove $\lim_{k\rightarrow \infty}\mathcal O_{\mathcal S_k} = \frac{9}{8}$ and then show that the multi-flow $\mathcal C^*_k$ constructed in our previous work is asymptotically optimal for $\mathcal P_{\mathcal S_k}$ and optimal if and only if $k = 1, 2, 6, 10$.

Citation: Kai Cai, Guangyue Han. An optimization approach to the Langberg-Médard multiple unicast conjecture. Advances in Mathematics of Communications, doi: 10.3934/amc.2021001
##### References:
 [1] K. Cai and G. Han, On network coding advantage for multiple unicast networks, in Proc. ISIT, (2015), 366–370. doi: 10.1109/ISIT.2015.7282478. [2] K. Cai and G. Han, Coding advantage in communications among peers, in Proc. ISIT, (2016), 2309–2313. doi: 10.1109/ISIT.2016.7541711. [3] K. Cai and G. Han, On the Langberg-Médard multiple unicast conjecture, Journal of Combinatorial Optimization, 34 (2017), 1114-1132.  doi: 10.1007/s10878-017-0132-2. [4] K. Cai and G. Han, On the Langberg-Médard $k$-unicast conjecture with k = 3, 4, in Proc. ISIT, (2018). [5] K. Cai and G. Han, The Langberg-Médard multiple unicast conjecture: Stable 3-pair networks, in Proc. ISIT, (2020). [6] N. Harvey, R. Kleinberg and A. Lehman, On the capacity of information networks, IEEE Trans. Inf. Theory, 52 (2006), 2345-2364.  doi: 10.1109/TIT.2006.874531. [7] K. Jain, V. V. Vazirani and G. Yuval, On the capacity of multiple unicast sessions in undirected graphs, IEEE/ACM Trans. Networking, 14 (2006), 2805-2809.  doi: 10.1109/TIT.2006.874543. [8] M. Langberg and M. M$\acute{e}$dard, On the multiple unicast network coding conjecture, in Proc. 47th Annual Allerton, (2009), 222–227. doi: 10.1109/ALLERTON.2009.5394800. [9] Z. Li and B. Li, Network coding: The case of multiple unicast sessions, in Proc. 42nd Annual Allerton, (2004). [10] Z. Li and C. Wu, Space information flow: Multiple unicast, in Proc. ISIT 2012, (2012), 1897–1901. doi: 10.1109/ISIT.2012.6283627. [11] J. Li, X. Chen, X. Sun, Z. Li, and Y. Yang, Quantum network coding for multi-unicast problem based on 2D and 3D cluster states, Science China-Information Sciences, 59 (2016), 042301. doi: 10.1007/s11432-016-5539-3. [12] A. Schrijver, Combinatorial Optimization, Springer-Verlag, 2003. [13] T. Xiahou, C. Wu, J. Huang and Z. Li, A geometric framework for investigating the multiple unicast network coding conjecture, in Proc. NetCod, (2012), 37–42. doi: 10.1109/NETCOD.2012.6261881. [14] G. Xu, X. Chen, J. Li, C. Wang, Y. Yang and Z. Li, Network coding for quantum cooperative multicast, Quantum Information Processing, 14 (2015), 4297-4322.  doi: 10.1007/s11128-015-1098-6. [15] Y. Yang, X. Yin, X. Chen, Y. Yang and Z. Li, A note on the multiple-unicast network coding conjecture, IEEE Communications Letters, 18 (2014), 869-872.  doi: 10.1109/LCOMM.2014.040214.140280. [16] R. Yeung, S.-Y. Li, and N. Cai, Network Coding Theory (Foundations and Trends in Communications and Information Theory), Now Publishers Inc., Hanover, MA, USA, 2006.

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##### References:
 [1] K. Cai and G. Han, On network coding advantage for multiple unicast networks, in Proc. ISIT, (2015), 366–370. doi: 10.1109/ISIT.2015.7282478. [2] K. Cai and G. Han, Coding advantage in communications among peers, in Proc. ISIT, (2016), 2309–2313. doi: 10.1109/ISIT.2016.7541711. [3] K. Cai and G. Han, On the Langberg-Médard multiple unicast conjecture, Journal of Combinatorial Optimization, 34 (2017), 1114-1132.  doi: 10.1007/s10878-017-0132-2. [4] K. Cai and G. Han, On the Langberg-Médard $k$-unicast conjecture with k = 3, 4, in Proc. ISIT, (2018). [5] K. Cai and G. Han, The Langberg-Médard multiple unicast conjecture: Stable 3-pair networks, in Proc. ISIT, (2020). [6] N. Harvey, R. Kleinberg and A. Lehman, On the capacity of information networks, IEEE Trans. Inf. Theory, 52 (2006), 2345-2364.  doi: 10.1109/TIT.2006.874531. [7] K. Jain, V. V. Vazirani and G. Yuval, On the capacity of multiple unicast sessions in undirected graphs, IEEE/ACM Trans. Networking, 14 (2006), 2805-2809.  doi: 10.1109/TIT.2006.874543. [8] M. Langberg and M. M$\acute{e}$dard, On the multiple unicast network coding conjecture, in Proc. 47th Annual Allerton, (2009), 222–227. doi: 10.1109/ALLERTON.2009.5394800. [9] Z. Li and B. Li, Network coding: The case of multiple unicast sessions, in Proc. 42nd Annual Allerton, (2004). [10] Z. Li and C. Wu, Space information flow: Multiple unicast, in Proc. ISIT 2012, (2012), 1897–1901. doi: 10.1109/ISIT.2012.6283627. [11] J. Li, X. Chen, X. Sun, Z. Li, and Y. Yang, Quantum network coding for multi-unicast problem based on 2D and 3D cluster states, Science China-Information Sciences, 59 (2016), 042301. doi: 10.1007/s11432-016-5539-3. [12] A. Schrijver, Combinatorial Optimization, Springer-Verlag, 2003. [13] T. Xiahou, C. Wu, J. Huang and Z. Li, A geometric framework for investigating the multiple unicast network coding conjecture, in Proc. NetCod, (2012), 37–42. doi: 10.1109/NETCOD.2012.6261881. [14] G. Xu, X. Chen, J. Li, C. Wang, Y. Yang and Z. Li, Network coding for quantum cooperative multicast, Quantum Information Processing, 14 (2015), 4297-4322.  doi: 10.1007/s11128-015-1098-6. [15] Y. Yang, X. Yin, X. Chen, Y. Yang and Z. Li, A note on the multiple-unicast network coding conjecture, IEEE Communications Letters, 18 (2014), 869-872.  doi: 10.1109/LCOMM.2014.040214.140280. [16] R. Yeung, S.-Y. Li, and N. Cai, Network Coding Theory (Foundations and Trends in Communications and Information Theory), Now Publishers Inc., Hanover, MA, USA, 2006.
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