Problem complexity vs. solution complexity

The previous two chapters focus on developing optimal polynomial-time solutions following formal optimization methods from operations research (OR). For some problems, such an approach may not be always effective, and could lead to nonpolynomial-time solutions. For these problems, a customized approach following algorithm design from computer science (CS) could be more effective and lead to a polynomial-time solution.

It is important to distinguish a (solution) algorithm's complexity from the underlying problem's complexity. A problem's complexity determines the potential complexity of any algorithm that is designed to solve this problem. That is, for a problem not in P, unless P = NP, any algorithm that can find an optimal solution to this problem must have nonpolynomial-time complexity. In contrast, if an algorithm (design to solve the problem) has a nonpolynomial-time complexity, we cannot claim that this problem is not in P. Another algorithm designed by someone else may well solve the problem with a polynomial-time complexity.

In this chapter, we illustrate the above approaches and ideas with a case study. This case study is concerned with an optimal relay node assignment problem that arises in cooperative communications (CC) [139]. We first formulate the problem as a mixed-integer linear programming (MILP), following OR's optimization approach.