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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...

## Context in source publication

**Context 1**

... 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. ...

## Similar publications

Constraint Satisfaction Problem (CSP) is a framework for modeling and solving a variety of real-world problems. Once the problem is expressed as a finite set of constraints, the goal is to find the variables' values satisfying them. Even though the problem is in general NP-complete, there are some approximation and practical techniques to tackle it...

## Citations

... 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). ...

Rank aggregation is widely used in group decision-making and many other applications where it is of interest to consolidate heterogeneous ordered lists. Oftentimes, these rankings may involve a large number of alternatives , contain ties, and/or be incomplete, all of which complicate the use of robust aggregation methods. In particular, these characteristics have limited the applicability of the aggregation framework based on the Kemeny-Snell distance, which satisfies key social choice properties that have been shown to engender improved decisions. This work introduces a binary programming formulation for the generalized Kemeny rank aggregation problem-whose ranking inputs may be complete and incomplete, with and without ties-and compare it to a modified version of a recently developed integer programming formulation for the generalized Kendall-tau distance. The new formulation has fewer variables and constraints, which leads to faster solution times. Moreover, we develop a new social choice property, the Non-strict Extended Condorcet Criterion, which can be regarded as a natural extension of the well-known Condorcet criterion and the Extended Con-dorcet criterion. Unlike its parent properties, the new property is adequate for handling complete rankings with ties. The property is leveraged to develop a structural decomposition algorithm, through which certain large instances of the NP-hard Kemeny rank aggregation problem can be solved exactly in a practical amount of time. To test the practical implications of the new formulation and social choice property, we work with instances constructed from a probabilistic distribution and benchmark instances from PrefLib, a library of preference data.

... 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. ...

We present a method, based on formulation symmetry, for generating Mixed-Integer Linear Programming (MILP) relaxations with fewer variables than the original symmetric MILP. Our technique also extends to convex MINLP, and some nonconvex MINLP with a special structure. We showcase the effectiveness of our relaxation when embedded in a decomposition method applied to two important applications (multi-activity shift scheduling and multiple knapsack problem), showing that it can improve CPU times by several orders of magnitude compared to pure MIP or CP approaches.

... The idea is to exploit symmetries to avoid the useless resolution of symmetrical µSCSPs. For that, we need to unify and extend the constraint approaches [Cohen et al., 2006] and the ones in GGP [Schiffel, 2010]. ...

The game description language with incomplete information (GDL-II) is expressive enough to capture partially observable stochastic multi-agent games. Unfortunately, such expressiveness does not come without a price: the problem of finding a winning strategy is NE XP NP -hard, a complexity class which is far beyond the reach of modern constraint solvers. In this paper, we identify a P SPACE -complete fragment of GDL-II, where agents share the same (partial) observations. We show that this fragment can be cast as a decomposable stochastic constraint satisfaction problem (SCSP) which, in turn, can be solved using general-purpose constraint programming techniques. Namely, we develop a constraint-based sequential decision algorithm for GDL-II games which exploits constraint propagation and Monte Carlo sampling based. Our algorithm, validated on a wide variety of games, significantly outperforms the state-of-the-art general game playing algorithms.

... Pour chaque µSCSP t dans {1, . . . , T }, [12, 16]), et constraint symmetries qui préservent l'ensemble des contraintes [4]. MAC-UCB-SYM exploite les symétries de contraintes, charactérisées par les automorphismes de la microstructure complémentaire du réseau. ...

Le game description language avec informations incomplètes (GDL-II) est assez expressif pour représenter les jeux stochastiques multi-agents avec observation partielle. Malheureusement, une telle expressivité n’est pas possible sans un prix : le problème consistant à trouver une stratégie gagnante est NExp^(NP)-hard, une classe de complexité qui est bien au-delà de la portée des solvers actuels. Dans ce papier, nous identifions un fragment Pspace-complete de GDL-II, où les agents partagent les mêmes observations (partielles). Nous montrons que ce fragment peut être encapsulé dans un problème de satisfaction de contraintes stochastiques décomposable (SCSP) qui, par tour, peut être résolu en utilisant des techniques de programmation par contraintes
usuelles. Dès lors, nous avons développé un algorithme de décisions séquentielles fondé sur les contraintes pour les jeux GDL-II exploitant la propagation par contraintes, l’évaluation Monte-Carlo et la détection de symétries. Notre algorithme, vérifié sur une large variété de jeux, surpasse aisément l’état de l’art des algorithmes du general game playing (GGP).

... Most previous research in this domain has concentrated on domain-filtering operations based on various forms of consistency: a value is removed from a domain if an algorithm running in low-order polynomial time demonstrates that this assignment cannot be part of a solution. Other reduction operations include the elimination of values by interchangeability or substitutability [7,11], the merging of domain values [10], the elimination of variables [9,5,3] and the introduction of symmetry-breaking constraints [1,8]. ...

Elimination of inconsistent values in instances of the constraint satisfaction problem (CSP) conserves all solutions. Elimination of substitutable values conserves at least one solution. We show that certain values which are neither inconsistent nor substitutable can also be deleted while conserving at least one solution. This allows us to state novel rules for the elimination of values in a binary CSP. From a practical point of view, we show that one such rule can be applied in the same asymptotic time complexity as neighbourhood substitution but is strictly stronger.
An alternative to the elimination of values from domains is the elimination of variables. We give novel satisfiability-preserving variable elimination operations. In each case we show that if the instance is satisfiable, then a solution to the original instance can always be recovered in low-order polynomial time from a solution to the reduced instance.

... 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[12]: 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 [1], 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 [14], 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. ...

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 [16] 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. ...

Edge matching puzzles have been amongst us for a long time now and traditionally they have been considered, both, a children’s game and an interesting mathematical divertimento. Their main characteristics have already been studied, and their worst-case complexity has been properly classified as a NP-complete problem. It is in recent times, specially after being used as the problem behind a money-prized contest, with a prize of 2US$ million for the first solver, that edge matching puzzles have attracted mainstream attention from wider audiences, including, of course, computer science people working on solving hard problems. We consider these competitions as an interesting opportunity to showcase SAT/CSP solving techniques when confronted to a real world problem to a broad audience, a part of the intrinsic, i.e. monetary, interest of such a contest. This article studies the NP-complete problem known as edge matching puzzle using SAT and CSP approaches for solving it. We will focus on providing, first and foremost, a theoretical framework, including a generalized definition of the problem. We will design and show algorithms for easy and fast problem instances generation, generators with easily tunable hardness. Afterwards we will provide with SAT and CSP models for the problems and we will study problem complexity, both typical case and worst-case complexity. We will also provide some specially crafted heuristics that result in a boost in solving time and study which is the effect of such heuristics.

... 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. ...

Many works have studied the properties of CSPs which are based on the structures of constraint networks, or based on the features of compatibility relations. Studies on structures rely generally on properties of graphs for binary CSPs and on properties of hypergraphs for the general case, that is CSPs with constraints of arbitrary arity. In the second case, us-ing the dual representation of hypergraphs, that is a refor-mulation of the instances, we can exploit notions and prop-erties of graphs. For the studies of compatibility relations, the exploitation of properties of graphs is possible studying a graph called microstructure which allows to reformulate in-stances of binary CSP. Unfortunately, this approach is limited to CSPs with binary constraints. In this paper, we propose theoretical tools based on graphs to represent microstructures for the general case. This approach avoids to exploit directly hypergraphs, even if the microstruc-ture based on hypergraphs has already been mentioned in (Cohen 2003). The advantage of such an approach is that the literature of Graph Theory is really more extended than one of Hypergraph Theory. Thus the theoretical results and effi-cient algorithms are more numerous, offering a larger number of existing tools which can be operated. We introduce here three possible definitions of microstructures based on graphs. We show how these representations can form new theoretical tools to generalize a number of results already obtained on binary CSPs. We think that these representations should be of interest for the community, firstly for the generalization of existing results, but also to obtain original results for CSPs with constraints of arbitrary arity.