We address the problem of verifying a constraint by a set of solutions S. This problem is present in almost all systems aiming at learning or acquiring constraints or constraint parameters. We propose an original approach based on MDDs. Indeed, the set of solutions can be represented by the MDD denoted by M DDS . Checking whether S satisfies a given constraint C can be done using M DD(C), the MDD that contains the set of solutions of C, and by searching if the intersection between M DD(S) and M DD(C) is equal to M DD(S). This step is equivalent to searching whether M DD(S) is included in M DD(C). Thus, we give an inclusion algorithm to speed up these calculations. Then, we generalize this approach for the computation of global constraint parameters satisfying C. Next, we introduce the notion of properties on the MDD nodes and define a new algorithm allowing to compute in only one step the set of parameters we are looking for. Finally, we present experimental results showing the interest of our approach.

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The product constraint ensures that the product of some variables will be greater than a given value, that is Π n i=1 xi ≥ w. With the emergence of stochastic problems, this constraint appears more and more frequently in practice. The variables are most often probability variables that represent the probability that an event will occur and the minimum bound is the minimum probability that must be satisfied. This is often done to guarantee a certain level of security or a certain quality of service. To deal with this constraint, it is tempting as proposed by many authors to take the logarithm of the sum and the bound in order to transform the product into a sum. In this article we show that this idea creates many problems and forbids an exact calculation. We propose and compare different representations allowing to compute the set of solutions of this problem exactly or up to a certain precision. We also give an efficient method to represent that constraint by a Multi-valued Decision Diagram (MDD) in order to combine this constraint with some others MDDs.

In robust optimization, finding a solution that solely respects the constraints is not enough. Usually, the uncertainty and unknown parameters of the model are represented by random variables. In such conditions, a good solution is a solution robust to most-likely assignments of these random variables. Recently, the Confidence constraint has been introduced by Mercier-Aubin et al. in order to enforce this type of robustness in constraint programming. Unfortunately, it is restricted to a conjunction of binary inequalities In this paper, we generalize the Confidence constraint to any constraint and propose an implementation based on Multi-valued Decision Diagrams (MDDs). The Confidence constraint is defined over a vector of random variables. For a given constraint C, and given a threshold, the Confidence constraint ensures that the probability for C to be satisfied by a sample of the random variables is greater than the threshold. We propose to use MDDs to represent the constraints on the random variables. MDDs are an efficient tool for representing combinatorial constraints, thanks to their exponential compression power. Here, both random and decision variables are stored in the MDD, and propagation rules are proposed for removing values of decision variables that cannot lead to robust solutions. Furthermore, for several constraints, we show that decision variables can be omitted from the MDD because lighter filtering algorithms are sufficient. This leads to gain an exponential factor in the MDD size. The experimental results obtained on a chemical deliveries problem in factories – where the chemicals consumption are uncertain – shows the efficiency of the proposed approach.