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Submodular Rate Region Models for Multicast Communication in Wireless Networks pp 105–153Cite as

Network Coding Bounds and Submodularity

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Part of the book series: Foundations in Signal Processing, Communications and Networking ((SIGNAL,volume 14))

Abstract

This chapter applies the multicast models for wireless networks, in particular, the proposed generalized submodular cut model and the proposed penalized polymatroid broadcast model, to general outer bounds, inner bounds, and approximations of the multicast capacity regions of discrete memoryless networks. It focuses on the class of independent noise discrete memoryless networks, i.e., the outputs are independent given the inputs. Determining the multicast capacity region of discrete memoryless networks, which includes the capacity (region) of point-to-point communication, bidirectional communication, and multiple access communication in relay networks, has been a longstanding open problem. The multicast capacity region has only recently been determined for some special cases, namely, graphical networks, hypergraphical networks, deterministic linear finite field networks, networks of independent deterministic broadcast channels, and networks of independent erasure broadcast channels with erasure location side-information at the destinations. For general networks, one has to resort to studying inner and outer bounds on the multicast capacity region. The analysis in this chapter focuses on four multicast rate regions related to the multicast capacity region, namely, the cut-set outer bound, the independent input approximation of the cut-set outer bound, the noisy network coding inner bound, and the elementary hypergraph decomposition inner bound. The goal of this chapter is to identify conditions under which these bounds admit representations by means of multicast rate regions that are generated by submodular cut rate regions, polymatroid broadcast rate regions, or hyperarc rate regions. In particular, this chapter reveals an intricate relationship between submodularity and independent noise networks, based on the submodularity of the entropy function on sets of random variables.

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Notes

  1. 1.

    Note that instead of defining the multi-message capacity region as a subset of \(\prod _{T\subset N}\mathscr {R}^N_+\), where \(r^T_a\) denotes the rate of node a for a multicast to T, we can equivalently define this capacity region as a hyperarc rate region \(\mathcal {G}\subset \mathscr {H}^N_+\) such that \(g_a(T)\) denotes the rate of source node a for a multicast to terminal set T. The transformation between both representations is done by appropriate reordering and grouping of the source rates into hyperarc rates and vice versa.

  2. 2.

    Incidentally, \(H(Y_A|Y_{A^{\mathsf {c}}}X_N)\) can be shown to be supermodular using the entropy chain rule and the submodularity of the entropy function.

  3. 3.

    The addition is with respect to the binary finite field, i.e., modulo two addition.

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  1. Fakultät für Elektrotechnik und Informationstechnik, Technische Universität München, Munich, Germany

    Maximilian Riemensberger

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Riemensberger, M. (2018). Network Coding Bounds and Submodularity. In: Submodular Rate Region Models for Multicast Communication in Wireless Networks. Foundations in Signal Processing, Communications and Networking, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-65232-0_4

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  • DOI: https://doi.org/10.1007/978-3-319-65232-0_4

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