Optimal Index Codes With Near-Extreme Rates

The min-rank of a digraph was shown to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this paper, the graphs and digraphs of near-extreme min-ranks are studied. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. In particular, it is shown that the decision problem whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time. In addition, a circuit-packing bound is revisited, and several families of digraphs, optimal with respect to this bound, whose min-ranks can be found in polynomial time, are presented.