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from Part I - Methods for Optimal Solutions
Published online by Cambridge University Press: 05 May 2014
When one door of happiness closes, another opens; but often we look so long at the closed door that we do not see the one which has opened for us.
Helen KellerProblem 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.
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