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Content may be subject to copyright. # The ten solutions of the 5-queens problem.

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We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solut...

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... solution is represented as a 4-clique in this graph, rather than as an independent set of size 4 in the binary nogood hypergraph. The automorphisms of this graph are that: the vertices within either clique can be permuted; the vertices in one clique can be swapped with those in the other; and the eight isolated vertices (representing unary nogoods) can be permuted; and we can also compose these The 5-queens problem has ten solutions, shown in Figure 3. These solutions are divided into two equivalence classes by the geometric symmetries of the chessboard; they transform any solution into another solution from the same equivalence class. ...

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... Considering incomplete-ranking inputs exacerbates these computational difficulties. When solving the problem via the standard branch and bound algorithm, incompleteness increases solution symmetry, which is defined as a permutation of the values of the variables that preserves the set of solutions (Cohen et al. 2005, Liberti 2008). This has the effect of slowing down pruning of nodes and, consequently, leads to a larger branch and bound tree (Sherali and Smith 2001). ...
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... Given a Mathematical Programming (MP) formulation, we distinguish the automorphism group of its solution set (called the solution group) and the group of variable symmetries fixing the formulation (called the formulation group). The latter is usually defined as the group of variable index permutations which keep the objective function invariant, the right-hand-side constraint vector invariant, and permutes the order of the constraints (Cohen et al., 2005; Margot, 2002). It is very easy to show that the formulation group is a subgroup of the solution group. ...
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... This section introduces some terminology and results about symmetry from the field of constraint programming. Firstly, we define a symmetry as a permutation on the assignment space which preserves satisfaction to the constraints: Definition 4. A symmetry S for a CSP (V, D, C) is a permutation on the assignment space S : (V → D) → (V → D) such that C(α) = C(S(α)). It follows directly from the definition that symmetries of a CSP Π form algebraic groups under the composition relation. ...
... Audeì a, l'´ etude des microstructures a ´ egalement montré son intérêt dans des domaines voisins. Par exemple, pour leprobì eme de comptage de solutions , ou encore l'´ etude des symétries dans les CSP binaires [7, 26]. Il est clair que la microstructure constitue un outil très utile pour l'´ etude théorique des CSP. ...
... The study of the microstructure has also shown its interest in other fields. For example, for the problem of counting the number of solutions , or for the study of symmetries in binary CSPs [15,16]. Thus, the microstructure appears as an interesting tool for the study of CSPs, or more precisely, for the theoretical study of CSPs. ...
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The CSP formalism has shown, for many years, its interest for the representation of numerous kinds of problems, and also often provide effective resolution methods in practice. This formalism has also provided a useful framework for the knowledge representation as well as to implement efficient methods for reasoning about knowledge. The data of a CSP are usually expressed in terms of a constraint network. This network is a (constraints) graph when the arity of the constraints is equal to two (binary constraints), or a (constraint) hypergraph in the case of constraints of arbitrary arity, which is generally the case for problems of real life. The study of the structural properties of these networks has made it possible to highlight certain properties, which led to the definition of new tractable classes, but in most cases, they have been defined for the restricted case of binary constraints. So, several representations by graphs have been proposed for the study of constraint hypergraphs to extend the known results to the binary case. Another approach, finer, is interested in the study of the microstructure of CSP, which is defined by graphs. This helped, offering a new theoretical framework to propose other tractable classes. In this paper, we propose to extend the notion of microstructure to any type of CSP. For this, we propose three kinds of graphs that can take into account the constraints of arbitrary arity. We show how these new theoretical tools can already provide a framework for developing new tractable classes for CSPs. We think that these new representations should be of interest for the community, firstly for the generalization of existing results, but also to obtain original results.
... Actually, for the particular case k = n the expected value is 1, so we cannot expect solvers to take profit of such symmetries for reducing the search space. One should look at more general notions of constraint and solution symmetries  in order to find symmetries that can help reduce the search space. Also, observe that by simply fixing the position and rotation of one of the tokens of the puzzle solution, we can discard the obvious symmetric solutions of every puzzle. ...
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... The study of the microstructure has also shown its interest in other fields. For example, for the problem of counting the number of solutions (Angelsmark and Jonsson 2003), or for the study of symmetries in binary CSPs (Cohen et al. 2006; Mears, de la Banda, and Wallace 2009). Thus, the microstructure appears as an interesting tool for the study of CSPs, or more precisely, for the theoretical study of CSPs. ...
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