Red-blue covering problems and the consecutive ones property

Set Cover problems are of core importance in many applications. In recent research, the ''red-blue variants'' where blue elements all need to be covered whereas red elements add further constraints on the optimality of a covering have received considerable interest. Application scenarios range from data mining to interference reduction in cellular networks. As a rule, these problem variants are computationally at least as hard as the original set cover problem. In this work we investigate whether and how the well-known consecutive ones property, restricting the structure of the input sets, makes the red-blue covering problems feasible. We explore a sharp border between polynomial-time solvability and NP-hardness for these problems.