Many multi-terminal communication networks, content delivery networks, cache networks, and distributed storage systems can be modeled as a broadcast network. An explicit characterization of the capacity region of the general network coding problem is one of the best known open problems in network information theory. A simple set of bounds that are often used in the literature to show that certain rate tuples are infeasible are based on the graph-theoretic notion of cut. The standard cut-set bounds, however, are known to be loose in general when there are multiple messages to be communicated in the network. This dissertation focuses on broadcast networks, for which the standard cut-set bounds are closely related to union as a specific set operation to combine different simple cuts of the network. A new set of explicit network coding bounds, which combine different simple cuts of the network via a variety of set operations (not just the union), are established via their connections to extremal inequalities for submodular functions. The tightness of these bounds are demonstrated via applications to combination networks. The tightness of generalized cut-set bounds has been further explored by studying the problem of "latency capacity region" for a broadcast channel. An implicit characterization of this region has been proved by Tian, where a rate splitting based scheme was shown to be optimal. However, the explicit characterization of this region was only available when the number of receivers are less than three. In this dissertation, a precise polyhedral description of this region for a symmetric broadcast channel with complete message set and arbitrary number of users has been established. It has been shown that a set of generalized cut-set bounds, characterizes the entire symmetrical multicast region. The achievability part is proved by showing that every maximum rate vector is feasible by using a successive encoding scheme. The framework for achievability strongly relies on polyhedral combinatorics and it can be useful in network information theory problems when a polyhedral description of a region is needed. Moreover, it is known that there is a direct relationship between network coding solution and characterization of entropy region. This dissertation, also studies the symmetric structures in network coding problems and their relation with symmetrical projections of entropy region and introduces new aspects of entropy inequalities. First, inequalities relating average joint entropies rather than entropies over individual subsets are studied. Second, the existence of non-Shannon type inequalities under partial symmetry is studied using the concepts of Shannon and non-Shannon groups. Finally, due to the relationship between linear entropic vectors and representability of integer polymatroids, construction of such vector has been discussed. Specifically, It is shown that representability of the particularly constructed matroid is a sufficient condition for integer polymatroids to be linearly representable over real numbers. Furthermore, it has been shown that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid.
Many multi-terminal communication networks, content delivery networks, cache networks, and distributed storage systems can be modeled as a broadcast network. An explicit characterization of the capacity region of the general network coding problem is one of the best known open problems in network information theory. A simple set of bounds that are often used in the literature to show that certain rate tuples are infeasible are based on the graph-theoretic notion of cut. The standard cut-set bounds, however, are known to be loose in general when there are multiple messages to be communicated in the network. This dissertation focuses on broadcast networks, for which the standard cut-set bounds are closely related to union as a specific set operation to combine different simple cuts of the network. A new set of explicit network coding bounds, which combine different simple cuts of the network via a variety of set operations (not just the union), are established via their connections to extremal inequalities for submodular functions. The tightness of these bounds are demonstrated via applications to combination networks. The tightness of generalized cut-set bounds has been further explored by studying the problem of "latency capacity region" for a broadcast channel. An implicit characterization of this region has been proved by Tian, where a rate splitting based scheme was shown to be optimal. However, the explicit characterization of this region was only available when the number of receivers are less than three. In this dissertation, a precise polyhedral description of this region for a symmetric broadcast channel with complete message set and arbitrary number of users has been established. It has been shown that a set of generalized cut-set bounds, characterizes the entire symmetrical multicast region. The achievability part is proved by showing that every maximum rate vector is feasible by using a successive encoding scheme. The framework for achievability strongly relies on polyhedral combinatorics and it can be useful in network information theory problems when a polyhedral description of a region is needed. Moreover, it is known that there is a direct relationship between network coding solution and characterization of entropy region. This dissertation, also studies the symmetric structures in network coding problems and their relation with symmetrical projections of entropy region and introduces new aspects of entropy inequalities. First, inequalities relating average joint entropies rather than entropies over individual subsets are studied. Second, the existence of non-Shannon type inequalities under partial symmetry is studied using the concepts of Shannon and non-Shannon groups. Finally, due to the relationship between linear entropic vectors and representability of integer polymatroids, construction of such vector has been discussed. Specifically, It is shown that representability of the particularly constructed matroid is a sufficient condition for integer polymatroids to be linearly representable over real numbers. Furthermore, it has been shown that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid.