Multi-Dimensional Models Facilitate Automatic Reformulation: The Case Study of the SET Game

In this paper we describe a reformulation strategy for solving multidimensional Constraint Satisfaction Problems (CSPs). This strategy operates by iteratively considering, in isolation, each one of the uni-dimensional constraints in the problem, and exploits the approximate symmetries induced by the selected constraint on the domains in order to enforce this constraint on the simplified problem. We use the game of SET, a combinatorial card game, as a toy problem to motivate our strategy and to explain and illustrate its operation. However, we believe that our approach is applicable to more complex domains of scientific and industrial importance, and deserves more thorough investigations in the future. Our approach sheds a new light on the dynamic reformulation of multidimensional CSPs. Importantly, it advocates that modeling tools for Constraint Programming should allow the user to specify the constraints directly on the attributes of the domain objects (i.e., variables and values) so that their multi-dimensionality can be exploited during problem solving

Finding robust solutions for constraint satisfaction problems with discrete and ordered domains by coverings

Constraint programming is a paradigm wherein relations between variables are stated in the form of constraints. Many real life problems come from uncertain and dynamic environments, where the initial constraints and domains may change during its execution. Thus, the solution found for the problem may become invalid. The search forrobustsolutions for constraint satisfaction problems (CSPs) has become an important issue in the ¿eld of constraint programming. In some cases, there exists knowledge about the uncertain and dynamic environment. In other cases, this information is unknown or hard to obtain. In this paper, we consider CSPs with discrete and ordered domains where changes only involve restrictions or expansions of domains or constraints. To this end, we model CSPs as weighted CSPs (WCSPs) by assigning weights to each valid tuple of the problem constraints and domains. The weight of each valid tuple is based on its distance from the borders of the space of valid tuples in the corresponding constraint/domain. This distance is estimated by a new concept introduced in this paper: coverings. Thus, the best solution for the modeled WCSP can be considered as a most robust solution for the original CSP according to these assumptionsThis work has been partially supported by the research projects TIN2010-20976-C02-01 (Min. de Ciencia e Innovacion, Spain) and P19/08 (Min. de Fomento, Spain-FEDER), and the fellowship program FPU.Climent Aunés, LI.; Wallace, RJ.; Salido Gregorio, MA.; Barber Sanchís, F. (2013). Finding robust solutions for constraint satisfaction problems with discrete and ordered domains by coverings. Artificial Intelligence Review. 1-26. https://doi.org/10.1007/s10462-013-9420-0S126Climent L, Salido M, Barber F (2011) Reformulating dynamic linear constraint satisfaction problems as weighted csps for searching robust solutions. 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