In this paper a two-stage algorithm for finding non- dominated subsets of partially ordered setsis established. A connection is then made with dimension reduction in time-dependentdynamic programming via the notion of a bounding label, a function that boundsthe state-transition cost functions. In this context, the computational burden is partitionedbetween a time-independent dynamic programming step carried out on the bounding label anda direct evaluation carried out on a subset of “real" valued decisions. A computationalapplication to time-dependent fuzzy dynamic programming is presented.

We consider a system of three queues and two types of packets. Each packet arriving at this system finds in front of it a controller who either sends it in the first queue or rejects it according to a QoS criterion. When the packet finishes its service in the first queue, it is probabilistically routed to one of two other parallel queues. The objective is to minimize a QoS discounted cost over an infinite horizon. The cost function is composed of a waiting cost per packet in each queue and a rejection cost in the first queue. Subsequently, we generalize this problem by considering a system of (m+1) queues and n types of packets. We show that an optimal policy is monotonic.

We consider a stochastic approach in order to define an equilibrium model for a traffic-network problem. In particular, we assume a Markovian behaviour of the users in their movements throughout the zones of the traffic area. This assumption turns out to beeffective at least in the context of urban traffic, where, in general, the users tend to travel by choosing the path they find more convenient and not necessarily depending on thealready travelled part.The developed model is a homogeneous Markov chain, whosestationarydistributions (if any) characterize the equilibrium.

This paper considers two backup schemes for a database system: a database is updated at a nonhomogeneous Poisson process and an amount of updated files accumulates additively. To ensure the safety of data, full backups are performed at time NT or when the total updated files have exceeded a threshold level K, and between them, cumulative backups as one of incremental backups are made at periodic times iT(i = 1,2,...,N - 1). Using the theory of cumulative processes, the expected cost is obtained, and an optimal number N* of cumulative backup and an optimal level K* of updated files which minimize it are analytically discussed. It is shown as examples that optimal number and level are numerically computed when two costs of backup schemes are given.

Cet article est le premier d'une série de deux articles oùnous présentons les principales caractéristiques d'un nouveauformalisme pour l'approximation polynomiale (algorithmiquepolynomiale à garanties de performances pour les problèmesNP-difficiles). Ce travail est l'occasion d'unregard critique sur ce domaine et de discussions sur la pertinencedes notions usuelles. Il est aussi l'occasion de se familiariseravec l'approximation polynomiale, de comprendre ses enjeux etses méthodes. Ces deux articles s'adressent donc autant auxspécialistes qu'aux non spécialistes de ce domaine. Nous insistons toutparticulièrementsur l'intérêt, tant théorique qu'opérationnel, de mettre enévidence une structure au sein de la classe NPO desproblèmes d'optimisation de NP. Dans ce premier article, nousnous intéressons aux outils qui permettent d'évaluer,dans l'absolu, les propriétés d'approximation de problèmesdifficiles. Nous discutons notamment les notions de chaînesd'approximation, de niveau d'approximation, d'ordre de difficultéainsi que deux notions de limites (par rapport à une suited'algorithmes et par rapport aux instances). Chaque notion estlargement discutée et illustrée par de nombreux exempleschoisis essentiellement pour leur valeur pédagogique.
Mots Clés. Complexité, difficulté intrinsèque, analyse des algorithmes et des problèmes,algorithmes d'approximation. Classification Mathématique. 68Q15, 68Q17, 68Q25, 68W25.