Eliminating an off-diagonal term from a design matrix brings value in a few different contexts. For a coupled design, it can decouple the design matrix. For a decoupled design, it will increase the flexibility of design interaction structure and thereby reducing the imaginary complexity. While eliminating an off-diagonal term is always preferred in the Axiomatic Design framework, the cost and benefit of eliminating it must be balanced to justify the effort. Assuming the cost is proportional to the number of off-diagonal terms to eliminate, this paper presents methods to identify a minimum set of off-diagonal terms that achieve the two objectives - decoupling and imaginary complexity. For the decoupling problem, the cycle matrix, C, provides an effective means to identify the optimal set of off-diagonal terms. The method we presented for the coupling problem guarantees the optimal solution. For the imaginary complexity problem, we presented a heuristic approach as a preliminary result to developing a complete method to select the optimal set.