A Categorical Approach to NP-Hard Optimization Problems

Aiming at developing a theoretical framework for the formal study of NP-hard optimization problems, which is built on precise mathematical foundations, we have focused on structural properties of optimization problems related to approximative issue. From the observation that, intuitively, there are many connections among categorical concepts and structural complexity notions, in this work we present a categorical approach to cope with some questions originally studied within Computational Complexity Theory. After defining the polynomial time soluble optimization problems category OPTS and the optimization problems category OPT, a comparison mechanism between them and an approximation system to each optimization problem have been introduced, following the basic idea of categorical shape theory. In this direction, we consider new insights and a deeper understanding of some basic questions inside the Structural Complexity field, by an universal language.