Generalized Lexicographic Products and the Index Coding Capacity

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.