This paper is the continuation of the paper “Autour
de nouvelles notions pour l'analyse des
algorithmes d'approximation: Formalisme unifié et classes
d'approximation” where a new formalism for polynomial
approximation and its basic tools allowing an “absolute”
(individual) evaluation the approximability properties of
NP-hard problems have been presented and discussed. In
order to be used for exhibiting a structure for the
class NPO (the optimization problems of NP),
these tools must be enriched with an “instrument” allowing
comparisons between approximability properties of different
problems (these comparisons must be independent on any specific
approximation result of the problems concerned). This instrument
is the approximability-preserving reductions. We show how to
integrate them in the formalism presented and propose the
definition of a new reduction unifying, under a specific point of
view a great number of existing ones. This new reduction allows
also to widen the use of the reductions, restricted until now
either between versions of a problem, or between problems, in
order to enhance structural relations between frameworks. They
allow, for example, to study how standard-approximation properties
of a problem transform into differential-approximation ones (for
the same problem, or for another one). Finally, we apply the
several concepts introduced to the study of the structure (and
hardness) of the instances of a problem. This point of view seems
particurarly fruitful for a better apprehension of the hardness of
certain problems and of the mechanisms for the design of efficient
solutions for them.