Abstract. Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state-of-the-art optimization solvers.Key words. systems of equations and inequalities, constraint programming, reformulation linearization technique, global constraint, interval arithmetic, safe rounding AMS subject classifications. 65H10, 65G40, 65H20, 93B18, 65G20 DOI. 10.1137/S00361429034361741. Introduction. Many applications in engineering sciences require finding all isolated solutions to systems of constraints over real numbers. These systems may be nonpolynomial and are difficult to solve: the inherent computational complexity is NP-hard and numerical issues are critical in practice (e.g., it is far from being obvious to guarantee correctness and completeness as well as to ensure termination). These systems, called numerical CSP (constraint satisfaction problem) in the rest of this paper, have been approached in the past by different interesting methods:
Discovering the set of closed frequent patterns is one of the fundamental problems in Data Mining. Recent Constraint Programming (CP) approaches for declarative itemset mining have proven their usefulness and flexibility. But the wide use of reified constraints in current CP approaches leads to difficulties in coping with high dimensional datasets. In this paper, we proposes the ClosedPattern global constraint to capture the closed frequent pattern mining problem without requiring reified constraints or extra variables. We present an algorithm to enforce domain consistency on ClosedPattern in polynomial time. The computational properties of this algorithm are analyzed and its practical effectiveness is experimentally evaluated.
Abstract. Sequential pattern mining (SPM) under gap constraint is a challenging task. Many efficient specialized methods have been developed but they are all suffering from a lack of genericity. The Constraint Programming (CP) approaches are not so effective because of the size of their encodings. In [7], we have proposed the global constraint PREFIX-PROJECTION for SPM which remedies to this drawback. However, this global constraint cannot be directly extended to support gap constraint. In this paper, we propose the global constraint GAP-SEQ enabling to handle SPM with or without gap constraint. GAP-SEQ relies on the principle of right pattern extensions. Experiments show that our approach clearly outperforms both CP approaches and the state-of-the-art cSpade method on large datasets.
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