Unicast-Uniprior Index Coding Problems: Minrank and Criticality

An index coding problem is called unicast-uniprior when each receiver demands a unique subset of messages while knowing another unique subset of messages apriori as side-information. In this work, an algorithm is given to compute the minrank of a unicast-uniprior problem. The proposed algorithm greatly simplifies the computation of minrank for unicast-uniprior problem settings, over the existing method whose complexity is exponential in the number of messages. First, some properties that are exclusive to the fitting matrix of a unicast-uniprior problem are established. These properties are then used to lay down the algorithm that computes the minrank.