Valeurs extrémales d'un problème d'optimisation combinatoire et approximation polynomiale

Abstract

As a subsequence of some of our previous works on complexity and polynomial approximation theory, we present some further reflections and arguments about extremal, optimal and worst, values (and solutions) of combinatorial optimization problems. This discussion leads us to consider a constant source of contradictions in complexity theory, the limits between constructibility and non-constructibility. In fact, complexity theory, in its current form, is founded on non-constructibility while, two of the main of its topics, the combinatorial optimization and the polynomial approximation theory need both a conceptual framework founded on constructibility. Approximation theory today goes beyond its framework of origin (a set of tools for finding fast solutions for NP-complete problems) since it strongly intervenes in the definition of new mathematical notions and objects making entirely part of the "arsenal" of complexity and it constitutes a major theoretical tool as well for understanding, deepening and enriching complexity theory as for better apprehending class NP. This recent "problemshift" for the polynomial approximation theory brings to the fore new and particularly interesting problems from both mathematical and epistemological points of view.