Yacine Boufkhad

New Encodings of Pseudo-Boolean Constraints into CNF

This paper answers affirmatively the open question of the existence of a polynomial size CNF encoding of pseudo-Boolean (PB) constraints such that generalized arc consistency (GAC) is maintained through unit propagation (UP). All previous encodings of PB constraints either did not allow UP to maintain GAC, or were of exponential size in the worst case. This paper presents an encoding that realizes both of the desired properties. From a theoretical point of view, this narrows the gap between the expressive power of clauses and the one of pseudo-Boolean constraints.

An upload bandwidth threshold for peer-to-peer Video-on-Demand scalability

We consider the fully distributed Video-on-Demand problem, where n nodes called boxes store a large set of videos and collaborate to serve simultaneously n videos or less between them. It is said to be scalable when Ω(n) videos can be distributively stored under the condition that any sequence of demands for these videos can always be satisfied. Our main result consists in establishing a threshold on the average upload bandwidth of a box, above which the system becomes scalable. We are thus interested in the normalized upload capacity u = upload bandwidth/video bitrate of a box. The number m of distinct videos stored in the system is called its catalog size. We show an upload capacity threshold of 1 for scalability in a homogeneous system, where all boxes have the same upload capacity. More precisely, a system with u ≪ 1 has constant catalog size m = O(1) (every box must store some data of every video). On the other hand, for u ≫ 1, an homogeneous system where all boxes have same upload capacity at least u admits a static allocation of m = Ω(n) videos into the boxes such that any adversarial sequence of video demands can be satisfied. Moreover, such an allocation can be obtained randomly with high probability. This result is generalized to a system of boxes that have heterogeneous upload capacities under some balancing conditions.

Non uniform selection of solutions for upper bounding the 3-SAT threshold

We give a new insight into the upper bounding of the 3-SAT threshold by the first moment method. The best criteria developed so far to select the solutions to be counted discriminate among neighboring solutions on the basis of uniform information about each individual free variable. What we mean by uniform information, is information which does not depend on the solution: e.g. the number of positive/negative occurrences of the considered variable. What is new in our approach is that we use non uniform information about variables. Thus we are able to make a more precise tuning, resulting in a slight improvement on upper bounding the 3-SAT threshold for various models of formulas defined by their distributions.