An overview on polynomial approximation of NP-hard problems

Vangelis Th. Paschos (1)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

The fact that it is very unlikely that a polynomial time algorithm could ever be devised for optimally solving NP-hard problems, strongly motivates both researchers and practitioners in trying to heuristically solving such problems, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find in reasonable time, not the best solution but one solution which is “near” the optimal one. Among classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in polynomial time by computing feasible solutions that are, under some predefined criterion, as near as possible to the optimal ones. The polynomial approximation theory deals with the study of such algorithms. This survey presents and analyzes in a first time approximation algorithms for some classical examples of NP-hard problems. In a second time, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what it is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.

Domaines: Complexité [cs.CC]; Recherche opérationnelle [math.OC]

Vangelis Th. Paschos. An overview on polynomial approximation of NP-hard problems. 2007. ⟨hal-00186549⟩