Adequate encodings for high-level constraints are a key ingredient for the application of SAT technology. In particular, cardinality constraints state that at most (at least, or exactly) k out of n propositional variables can be true. They are crucial in many applications. Although sophisticated encodings for cardinality constraints exist, it is well known that for small n and k straightforward...

We show that the satisfiability problem for quantifier-free theory of product structures with equicardinality relation is in NP. As an application, we extend combinatory array logic fragment to handle cardinality constraints. The resulting independent base element and index set theories.

Constraint programming (CP) is mainly based on filtering algorithms; their association with global constraints is one of the main strengths of CP. This chapter is an overview of these two techniques. Some of the most frequently used global constraints are presented. In addition, the filtering algorithms establishing arc consistency for two useful constraints, the alldiff and the global cardinal...

Weighted First-Order Model Counting (WFOMC) computes the weighted sum of models a first-order logic theory on given finite domain. Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called liftable. We introduce concept lifted interpretations as tool for formulating closed forms WFOMC. Using interpretations, we reconstruct closed-form formula FOMC in universally quanti...

Portfolio Selection: How to Integrate Complex Constraints For the standard Mean-Variance model for portfolio selection with linear constraints, there are several algorithms that can efficiently compute both a single point on the Pareto front and even the whole front. Unfortunately, commonly used constraints (e.g. cardinality constraints or buy-in thresholds) result in the optimization problem t...

Constraint programming (CP) is mainly based on filtering algorithms; their association with global constraints is one of the main strengths of CP. This chapter is an overview of these two techniques. Some of the most frequently used global constraints are presented. In addition, the filtering algorithms establishing arc consistency for two useful constraints, the alldiff and the global cardinal...

The SEt Reasoning Facility (SERF) integrates mechanisms for propagating membership propositions, deriving relations between sets, and reasoning about closure and cardinality into an efficient utility package for reasoning about sets . Assertions about relations between sets are compiled into a constraint network defined entirely in terms of union, complement, and emptiness constraints . The con...

We explore the criteria that contribute to the validity of modeling structures within the entity-relationship (ER) diagram. Our approach examines cardinality constraints in conjunction with the degree of the relationship to address constraint consistency and role uniqueness issues. Here we use both maximum and minimum cardinality constraints yielding a complete analysis of the structural validi...