The non-uniform demand network coding problem is posed as a single-source and multiple-sink network transmission problem where the sinks may have heterogeneous demands. In contrast with multicast problems, non-uniform demand problems are concerned with the amounts of data received by each sink, rather than the specifics of the received data. In this work, we enumerate non-uniform network demand scenarios under which network coding solutions can be found in polynomial time. This is accomplished by relating the demand problem with the graph coloring problem, and then applying results from the strong perfect graph theorem to identify coloring problems which can be solved in polynomial time. This characterization of efficiently-solvable non-uniform demand problems is an important step in understanding such problems, as it allows us to better understand situations under which the NP-complete problem might be tractable.
We briefly mention some results regarding the multiple multicast connections problem, since achievable solutions for such problems are also achievable for the non-uniform demand problem with the same demanded rates. (Of course, the reverse is not always true.) Many of these results consider the case of two sinks. @cite_10 , after enumeration of all possible scenarios, the authors conclude that in the case of two sinks with differing rates, linear coding is sufficient. The same conclusion is made in @cite_3 , although the authors use a different approach which considers a path-based enumeration. A characterization of the achievable data rate region using network coding is given for the two sink case. For more than two sinks, conditions under which solutions exist for the multiple connections problem have not been enumerated.
@cite_10 @cite_3
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Network coding shows that the data rate can be increased if information is allowed to be encoded in the network nodes. The recent work of S.-Y.R. (see IEEE Trans. Inform. Theory, vol.49, no.2, p.371-81, 2003) and R. Koetter and M. Medard (see Proc. INFOCOM, 2002) shows that linear network coding is sufficient for a single source multicast network. The restrictiveness of the use of linear code is still an unknown in the general multisource multicast network. We characterize the achievable information rate region for a single source node multi-source multicast network with two sinks. We further show that linear coding is sufficient for achieving the maximum network capacity. In this paper, we present an achievable rate region for double-unicast networks by assuming that the intermediate nodes perform random linear network coding, and the source and sink nodes optimize their strategies to maximize the achievable region. Such a setup can be modeled as a deterministic interference channel, whose capacity region is known. For the particular class of linear deterministic interference channels of our interest, in which the outputs and interference are linear deterministic functions of the inputs, we show that the known capacity region can be achieved by linear strategies. As a result, for a given set of network coding coefficients chosen by the intermediate nodes, the proposed linear precoding and decoding for the source and sink nodes will give the maximum achievable rate region for double-unicast networks. We further derive a suboptimal but easy-to-compute rate region that is independent of the network coding coefficients used at the intermediate nodes, and is instead specified by the min-cuts of the network. It is found that even this suboptimal region is strictly larger than the existing achievable rate regions in the literature. One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions For a network with one sender, n receivers (users) and m possible messages (files), caching side information at the users allows to satisfy arbitrary simultaneous demands by sending a common (multicast) coded message. In the worst-case demand setting, explicit deterministic and random caching strategies and explicit linear coding schemes have been shown to be order optimal. In this work, we consider the same scenario where the user demands are random i.i.d., according to a Zipf popularity distribution. In this case, we pose the problem in terms of the minimum average number of equivalent message transmissions. We present a novel decentralized random caching placement and a coded delivery scheme which are shown to achieve order-optimal performance. As a matter of fact, this is the first order-optimal result for the caching and coded multicasting problem in the case of random demands.