Structural Characteristics of Two-Sender Index Coding

This paper studies index coding with two senders. In this setup, source messages are distributed among the senders (possibly with overlapping of messages). In addition, there are multiple receivers, where each receiver having some messages a priori, known as side-information, is requesting one unique message such that each message is requested by only one receiver. Index coding in this setup is called two-sender unicast index coding (TSUIC). The main goal is to find the shortest aggregate encoded messages while allowing all receivers to decode their requested messages. Firstly, we form three independent sub-problems of a TSUIC problem based on whether the requested messages by receivers of those sub-problems are available only in one of the senders or in both senders. Then we express the optimal broadcast rate (the shortest normalized codelength) of the TSUIC problem as a function of the optimal broadcast rates of those independent sub-problems. In this way, we devise the structural characteristics of TSUIC. For the proofs of our results, we apply the notion of confusion graphs and a code-forming technique. To color a confusion graph in TSUIC, we introduce a new graph-coloring approach (different from the normal graph coloring), called two-sender graph coloring, and propose a way of grouping the vertices to analyze the number of colors used. Finally, we determine a class of TSUIC instances where a certain type of side-information can be removed without affecting their optimal broadcast rates.