In the present work we are interested in to provide a universal language for supporting formalisms to specify the approximation hierarchy system for an abstract NP-hard optimization problem. This work grew from the idea of providing a categorical view of structural complexity to optimization problems. The direction is aimed towards actually exploring the connections among the structural complexity aspects and categorical concepts, which may be viewed in a high-level, in a structuralistic sense. After introducing the optimization problems categories OPTS and OPT, as well as related questions, a formal system modelling the approximation hierarchy of a given optimization problem is provided, based on categorical shape theory.
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