Approximation Problems Categories

The notion of approximation problems was formally introduced by Johnson [3] in his pioneering paper on the approximation of combinatorial optimization problems, and it was also suggested a possible classification of optimization problems on grounds of their approximability properties. Since then, it was clear that, even though the decision versions of most NP-hard optimization problems are polynomial-time reducible to each other, they do not share the same approximability properties. In spite of some remarkable attempts, according to Ausiello [1] the reasons that a problem is approximable or nonapproximable are still unknown. The different behaviour of NP-hard optimization problems with respect to their approximability properties is captured by means of the definition of approximation classes and, under the “P = NP” conjecture, these classes form a strict hierarchy whose levels correspond to different degrees of approximation. In this paper we continue along the same line of research started in [4], towards to providing a categorical view of structural complexity to optimization problems. The main aim is to provide a universal language for supporting formalisms to specify the hierarchy approximation system for an abstract NP-hard optimization problem, in a general sense. From the observation that, intuitively, there are many connections among categorical concepts and structural complexity notions, we started defining two categories: the OPTS category of polynomial time soluble optimization problems, which morphisms are reductions, and the OPT category of optimization problems, having approximationpreserving reductions as morphisms. The study of approximation implies to create means of comparing optimization problems. The basic idea of approximation by models is a recurrent one in mathematics and in this direction a comparison mechanism between the OPTS and OPT categories has been introduced in [5]. In order to establish a formal ground for the study of the approximation properties of optimization problems, a system approximation to each optimization problem is constructed, based on categorical shape theory. In so doing, we were very much inspired in previous works