The index coding problem studies the fundamental limit on broadcasting multiple messages to their respective receivers with different sets of side information that are represented by a directed graph. The generalized lexicographic product structure in the side information graph is introduced as a natural condition under which the corresponding index coding problem can be decomposed into multiple interacting subproblems, each consisting of vertices with the same adjacency pattern with respect to other subproblems. For side information graphs with this structure, the capacity region is characterized in terms of the subproblem capacity regions combined in the same product structure. The proof is based on dual uses of random coding-one for a new multiletter characterization of the capacity region of a general index coding problem via joint typicality decoding and the other for a construction of a new multiletter code of matching rates from a single-letter code via joint typicality encoding. Several special cases are discussed that recover and strengthen known structural properties of the index coding capacity region.

In index coding, a server broadcasts multiple messages to their respective receivers, each with some side information that can be utilized to reduce the amount of communication from the server. Distributed index coding is an extension of index coding in which the messages are broadcast from multiple servers, each storing different subsets of the messages. In this paper, the optimal tradeoff among the message rates and the server broadcast rates, which is defined formally as the capacity region, is studied for a general distributed index coding problem. Inner and outer bounds on the capacity region are established that have matching sum-rates for all 218 non-isomorphic four-message problems with equal link capacities for all the links from servers to receivers. The proposed inner bound is built on a distributed composite coding scheme that outperforms the existing schemes by incorporating more flexible decoding configurations and enhanced fractional rate allocations into two-stage composite coding, a scheme that was originally introduced for centralized index coding. The proposed outer bound is built on the polymatroidal axioms of entropy, as well as functional dependences such as the fd-separation introduced by the multi-server nature of the problem. This outer bound utilizes general groupings of servers with different levels of granularity, which allows a natural tradeoff between computational complexity and tightness of the bound, and includes and improves upon all existing outer bounds for distributed index coding. Specific features of the proposed inner and outer bounds are demonstrated through concrete examples with four or five messages.

Simplification methods are introduced for composite coding, which is an existing layered random coding technique for the index coding problem. As the problem size grows, the original number of composite indices grows exponentially and the number of possible decoding configurations (decoding sets) grows super exponentially, leading to considerably high computational complexity. The proposed simplifications address both issues and do not affect the performance (tightness) of the coding scheme. Removing composite indices is achieved by pairwise comparison of any two indices and removing one if its corresponding rate can be transferred without loss to the other in the expressions of the achievable rate region. Decoding configurations are reduced by establishing a baseline or natural decoding configuration, where no smaller decoding configuration can provide a strictly larger rate region. A heuristic method is also proposed for reducing the number of composite indices even further, but possibly with some performance loss. Numerical results demonstrate good performance with substantial reduction in complexity. To achieve the capacity region for all 9608 non-isomorphic index coding problems with n = 5, a single natural decoding configuration per problem and less than 3 out of 25-1=31 composite indices are sufficient, on average. In only 31 problems, 7 to at most 10 composite indices are used.

The distributed index coding problem is studied, whereby multiple messages are stored at different servers to be broadcast to receivers with side information. First, the existing composite coding scheme is enhanced for the centralized (single-server) index coding problem, which is then merged with fractional partitioning of servers to yield a new coding scheme for distributed index coding. New outer bounds on the capacity region are also established. For all distributed index coding problems with n ≤ 4 messages and equal server link capacities, the achievable sum-rate of the proposed distributed composite coding scheme match the outer bounds, thus establishing the sum-capacity for these problems.

For a class of index coding problems with side information graph having the Ramsey number R(i, j) upper bounded by ciajb, it is shown that the clique covering scheme approximates the broadcast rate within a multiplicative factor of O(na+b/a+b+1), where n is the number of messages. Based on this result and known bounds on Ramsey numbers, it is demonstrated that the broadcast rate of planar graphs, line graphs, and fuzzy circular interval graphs can be approximated within a factor of (2n)2/3.

In this paper, we study the capacity region of the general distributed index coding. In contrast to the traditional centralized index coding where a single server contains all n messages requested by the receivers, in the distributed index coding there are 2n- 1 servers, each containing a unique non-empty subset J of the messages and each is connected to all receivers via a noiseless independent broadcast link with an arbitrary capacity CJ≥ 0. First, we generalize the existing outer bound on the capacity region of the centralized problem to the distributed case. Next, building upon the existing centralized composite coding scheme, we propose three distributed composite coding schemes and derive the corresponding inner bounds on the capacity region. We present a number of interesting numerical examples, which highlight the subtleties and challenges of dealing with the distributed index coding, even for very small problem sizes of n = 3 and n = 4.

The capacity region of the index coding problem is characterized through the notion of confusion graph and its fractional chromatic number. Based on this multiletter characterization, several structural properties of the capacity region are established, some of which are already noted by Tahmasbi, Shahrasbi, and Gohari, but proved here with simple and more direct graph-theoretic arguments. In particular, the capacity region of a given index coding problem is shown to be simple functionals of the capacity regions of smaller subproblems when the interaction between the subproblems is none, one-way, or complete.

A series of extensions of the index coding schemes based on time sharing by Birk and Kol, by Blasiak, Kleinberg, and Lubetzky, and by Shanmugam, Dimakis, and Langberg are presented. Each extension strictly improves upon the previous extensions as well as the existing schemes. The main idea behind these extensions is local time sharing over subproblems introduced by Shanmugam et al., in which the local side information available at each receiver is exploited to send the subproblm indices with a fewer number of transmissions. The final extension, despite being the best in this class of coding schemes, is shown to be still suboptimal, characterizing the fundamental limit of local time sharing.

A new inner bound on the capacity region of the general index coding problem is established. Unlike most existing bounds that are based on graph theoretic or algebraic tools, the bound relies on a random coding scheme and optimal decoding, and has a simple polymatroidal single-letter expression. The utility of the inner bound is demonstrated by examples that include the capacity region for all index coding problems with up to five messages (there are 9846 nonisomorphic ones).