Abstract
Constructing k independent sessions between k source-sink pairs with the help of a linear operation at each vertex is one of the most standard problems in network coding. For an unbounded k, this is known to be NP-hard. Very recently, a polynomial-time algorithm was given for k = 2 [Wang and Shroff, ISIT 2007], but was open for a general (constant) k. This paper gives a polynomial-time algorithm for this problem under the assumption that the size of the finite field for the linear operations is bounded by a fixed constant.
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References
Adler, M., Harvey, N.J., Jain, K., Kleinberg, R.D., Lehman, A.R.: On the capacity of information networks. In: Proc. 17th ACM-SIAM SODA, pp. 241–250 (2006)
Ahlswede, R., Cai, N., Li, S.-Y.R., Yeung, R.W.: Network information flow. IEEE Transactions on Information Theory 46, 1204–1216 (2000)
Dougherty, R., Freiling, C., Zeger, K.: Insufficiency of linear coding in network information flow. IEEE Transactions on Information Theory 51, 2745–2759 (2005)
Dougherty, R., Zeger, K.: Nonreversibility and equivalent constructions of multiple-unicast networks. IEEE Transactions on Information Theory 52, 5067–5077 (2006)
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121 (1980)
Fragouli, C., Soljanin, E.: Information flow decomposition for network coding. IEEE Transactions on Information Theory 52, 829–848 (2006)
Harvey, N.J., Karger, D.R., Murota, K.: Deterministic network coding by matrix completion. In: Proc. 16th ACM-SIAM SODA, pp. 489–498 (2005)
Harvey, N.J., Kleinberg, R.D., Lehman, A.R.: Comparing network coding with multicommodity flow for the k-pairs communication problem. MIT LCS Technical Report 964 (September 2004)
Ho, T., Karger, D.R., Médard, M., Koetter, R.: Network coding from a network flow perspective. In: Proc. IEEE International Symposium on Information Theory (2003)
Jaggi, S., Sanders, P., Chou, P.A., Effros, M., Egner, S., Jain, K., Tolhuizen, L.M.G.M.: Polynomial time algorithms for multicast network code construction. IEEE Transactions on Information Theory 51, 1973–1982 (2005)
Koetter, R.: Network coding home page, http://tesla.csl.uiuc.edu/~koetter/NWC/
Koetter, R., Médard, M.: Beyond routing: An algebraic approach to network coding. In: Proc. 21st Annual Joint Conference of the IEEE Computer and Communications Societies, pp. 122–130 (2002)
Langberg, M., Sprintson, A., Bruck, J.: Network coding: A computational perspective. In: Proc. 40th Conference on Information Sciences and Systems (2006)
Langberg, M., Sprintson, A., Bruck, J.: The encoding complexity of network coding. IEEE Transactions on Information Theory 52, 2386–2397 (2006)
Lehman, A.R.: Network Coding. PhD thesis. MIT, Cambridge (2005)
Lehman, A.R., Lehman, E.: Complexity classification of network information flow problems. In: Proc. 15th ACM-SIAM SODA, pp. 142–150 (2004)
Lehman, A.R., Lehman, E.: Network coding: Does the model need tuning? In: Proc. 16th ACM-SIAM SODA, pp. 499–504 (2005)
Li, S.-Y.R., Yeung, R.W., Cai, N.: Linear network coding. IEEE Transactions on Information Theory 49, 371–381 (2003)
Médard, M., Effros, M., Ho, T., Karger, D.: On coding for non-multicast networks. In: Proc. 41st Annual Allerton Conference on Communication, Control and Computing (2003)
Slivkins, A.: Parameterized tractability of edge-disjoint paths on directed acyclic graphs. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 482–493. Springer, Heidelberg (2003)
Tavory, A., Feder, M., Ron, D.: Bounds on linear codes for network multicast. ECCC Technical Report 33 (2003)
Wang, C.-C., Shroff, N.B.: Beyond the butterfly – A graph-theoretic characterization of the feasibility of network coding with two simple unicast sessions. In: Proc. IEEE International Symposium on Information Theory (2007)
Wang, C.-C., Shroff, N.B.: Intersession network coding for two simple multicast sessions. In: Proc. 45th Annual Allerton Conference on Communication, Control and Computing (2007)
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Iwama, K., Nishimura, H., Paterson, M., Raymond, R., Yamashita, S. (2008). Polynomial-Time Construction of Linear Network Coding. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_23
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DOI: https://doi.org/10.1007/978-3-540-70575-8_23
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