In the paper, the problem of the genome mapping of DNA molecules, is presented. In particular, the new approach — the Simplified Partial Digest Problem (SPDP), is analyzed. This approach, although easy in laboratory implementation and robust with respect to measurement errors, when formulated in terms of a combinatorial search problem, is proved to be strongly NP-hard for the general error-free case. For a subproblem of the SPDP, a simple O(nlogn)-time algorithm is given, where n is a number of restriction sites.

The multiparametric min max 0-1-Integer Programming (0-1-IP) problem relative to the objective function is a family of min max 0-1-IP problems which are related by having identical constraint matrix and right-hand-side vector. In this paper we present an algorithm to perform a complete multiparametric analysis relative to the objective function.

The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space Y are introduced. Under quasiconvexity assumptions we show that solutions of generalized variational inequality and of the primitive optimization problem are equivalent. Finally, we discuss the possibility to generalize the scheme of this paper to get further applications in vector optimization.

Le système AutoGraphiX (AGX1 et AGX2) permet, parmi d'autres fonctions, la génération automatique de conjectures en théorie des graphes et, dans une version plus récente, la preuve automatique de conjectures simples. Afin d'illustrer ces fonctions et le type de résultats obtenus, nous étudions systématiquement ici des conjectures obtenues par ce système et de la forme $\underline{b}_{n} \, \le \, g \, \oplus \, i \, \le \, \overline{b}_{n}$ où g désigne la maille (ou longueur du plus petit cycle) du graphe G=(V, E), i un autre invariant choisi parmi le nombre de stabilité, le rayon, le diamètre, le degré minimum, moyen ou maximum, $\underline{b}_{n} $ et $ \overline{b}_{n} $ des fonctions de l'ordre n = |V| de G les meilleures possibles, enfin $ \oplus $ correspond à une des opérations +,-,×,/. 48 telles conjectures sont obtenues: les plus simples sont démontrées automatiquement et les autres à la main. De plus 12 autres conjectures ouvertes et non encore étudiées sont soumises aux lecteurs.