This paper is devoted to an oriented coloring problem motivated by a task assignment model. A recent result established the NPcompleteness of deciding whether a digraph is k-oriented colorable; we extend this result to the classes of bipartite digraphs and circuit-free digraphs. Finally, we investigate the approximation of this problem: both positive and negative results are devised.

Given an instance I of an optimization constraint satisfaction problem (CSP), finding solutions with value at least the expected value of a random solution is easy. We wonder how good such solutions can be. Namely, we initiate the study of ratio ρE(I) = (EX [v(I, X)] − wor(I))/(opt(I) − wor(I)) where opt(I), wor(I) and EX [v(I, X)] refer to respectively the optimal, the worst, and the average solution values on I. We here focus on the case when the variables have a domain of size q ≥ 2 and the constraint arity is at most k ≥ 2, where k, q are two constant integers. Connecting this ratio to the highest frequency in orthogonal arrays with specified parameters, we prove that it is Ω(1/n k/2) if q = 2, Ω(1/n k−1− log p κ (k−1)) where p κ is the smallest prime power such that p κ ≥ q otherwise, and Ω(1/q k) in (max{q, k} + 1})-partite instances.

RésuméUn intervalle X d'un tournoi T est un ensemble de sommets de T tel que tout sommet extérieurà X domine ou est dominé par tous les sommets de X. Nous caractérisons les tournois dont tous les intervalles acycliques non vides sont des singletons et qui sont critiques pour cette propriété, c'est-à-dire que la suppression d'un sommet quelconque du tournoi donne naissanceà au moins un intervalle acyclique de plus de 2 sommets. Ces tournois sont exactement ceux construits comme la composition d'un tournoi quelconque avec des tournois circulants. Ce travail sur les intervalles acycliques aété motivé par la recherche de structures ordonnées dans des tournois pour lesquels aucun ordre médian ne s'impose naturellement. Pour citer cet article : J.F. Culus, B. Jouve, C. R. Acad. Sci. Paris, Ser. I 340 (2005). AbstractTournaments without acyclic interval. An interval X of a tournament T is a vertex subset of T such that any vertex not in X either dominates or is dominated by all of the vertices in X. We caracterize the tournaments such that the only non empty acyclic intervals are the singletons and which are critical for that property, that is whenever a vertex is removed at least one acyclic interval with more than 2 vertices is created. These tournaments are exactly those which are the composition of any tournament with circulant tournaments. That work on acyclic intervals was motivated by the study of tournaments for which no median order forced itself naturally.

No abstract

We introduce the convex circuit-free coloration and convex circuit-free chromatic number − → χ a (− → G) of an oriented graph − → G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χ a (− → G) ≤ k is NP-complete if k ≥ 3 and polynomial if k ≤ 2. We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and characterize the tournaments that are vertex-critical for the convex circuit-free coloration.