A unified formulation and approach to the analysis of approximation algorithms: the structure of NP optimisation problems

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 of 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 particularly fruitful for a better apprehension of the hardness of certain problems and of the mechanisms for the design of efficient solutions for them.