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The concept of flexibility -originated in the context of heat exchanger networks- is associated with a substructure which guarantees the performance of the original structure, in a given range of possible states. We extend this concept to combinatorial optimization problems, and prove several computational complexity results in this new framework. Under some monotonicity conditions, we prove that a combinatorial optimization problem polynomially transforms to its associated flexibility problem, but that the converse need not be true. In order to obtain polynomial flexibility problems, we have to restrict ourselves to combinatorial optimization problems on matroids. We also prove that, when relaxing in different ways the matroid structure, the flexibility problems become NP-complete. This fact is shown by proving the NP -completeness of the flexibility problems associated with the Shortest Path, Minimum Cut and Weighted Matching problems.