We study the weighted circuit constraint in the context of constraint programming. It appears as a substructure in many practical applications, particularly routing problems. We propose a domain filtering algorithm for the weighted circuit constraint that is based on the 1-tree relaxation of Held and Karp. In addition, we study domain filtering based on an additive bounding procedure that combines the 1-tree relaxation with the assignment problem relaxation. Experimental results on Traveling Salesman Problem instances demonstrate that our filtering algorithms can dramatically reduce the problem size. In particular, the search tree size and solving time can be reduced by several orders of magnitude, compared to existing constraint programming approaches. Moreover, for medium-size problem instances, our method is competitive with the state-of-the-art special-purpose TSP solver Concorde.
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:
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