On the satisfiability of symmetrical constrained satisfaction problems

Puget, Jean‐François

doi:10.1007/3-540-56804-2_33

Cited by 99 publications

(108 citation statements)

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“…As far back as 1874 an article proposed making use of symmetries in the eight queens problem [29]. This subject has been addressed by a number of researchers in the past decades, particularly in the constraint programming community (see for example [28,30,31,32,33,34,35,36,37]). Note that there are several definitions of symmetry which differ in the attributes they apply to (only the values, only the variables, or variable-value pairs) and in what they preserve (the constraints or the set of solutions) [36].…”

Section: Definition 9 Let S Be a Subset Of ω Given A Subset S Of Smentioning

confidence: 99%

Dominance rules in combinatorial optimization problems

Jouglet

Carlier

2011

European Journal of Operational Research

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The aim of this paper is to study the concept of a "dominance rule" in the context of combinatorial optimization. A dominance rule is established in order to reduce the solution space of a problem by adding new constraints to it, either in a procedure that aims to reduce the domains of variables, or directly in building interesting solutions. Dominance rules have been extensively used over the last fifty years. Surprisingly, to our knowledge, no detailed description of them can be found in the literature other than a few short formal descriptions in the context of enumerative methods. We are therefore proposing an investigation into what dominance rules are. We first provide a definition of a dominance rule with its different nuances. Next, we analyze how dominance rules are generally formulated and what are the consequences of such formulations. Finally, we enumerate the common characteristics of dominance rules encountered in the literature and in the usual process of solving combinatorial optimization problems.

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Section: Definition 9 Let S Be a Subset Of ω Given A Subset S Of Smentioning

confidence: 99%

Dominance rules in combinatorial optimization problems

Jouglet

Carlier

2011

European Journal of Operational Research

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“…One of the most successful techniques for handling symmetry in both SAT and general CSPs originates from the work of Puget [46], who showed that symmetries can be broken by adding one lexicographic ordering constraint per symmetry. Crawford et al [10] showed how this can be done by adding a set of simple "lex-constraints" or symmetry breaking predicates (SBPs) to the input specification to weed out all but the lexically-first solutions.…”

Section: Related Workmentioning

confidence: 99%

SymChaff: exploiting symmetry in a structure-aware satisfiability solver

Sabharwal

2008

Constraints

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This article presents a new low-overhead framework for representing and utilizing problem symmetry in propositional satisfiability algorithms. While many previous approaches have focused on symmetry extraction as a key component, the novelty in the proposed strategy lies in using high level problem description to pass on symmetry information to the SAT solver in a simple and concise form, in addition to the usual CNF formula. This information, comprising of the so-called symmetry sets and variable classes, captures variable semantics relevant to "complete multi-class symmetry" and is utilized dynamically to prune the search space. This allows us to address many limitations of alternative approaches like symmetry breaking predicates, implicit pseudo-Boolean representations, general group-theoretic methods, and ZBDDs. We demonstrate the efficacy of our technique through a solver called SymChaff, which achieves exponential speedup over DPLL-based SAT solvers on problems from both theory and practice, often by simply using natural tags or annotation in the problem specification.

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“…Such formula evaluates to true if and only if a mapping T from tuples in R to tuples in G (25) exists, such that it is total (26), mono-valued (i.e., a function) (27), and injective (28). Since such a function exists if and only if the size of its domain is less than or equal to the size of its codomain, β R,G SB (R, G, B) evaluates to true if and only if |R| ≤ |G|.…”

Section: Figmentioning

confidence: 99%

“…To this end, many techniques have been proposed, in order to modify (or reformulate) the given constraint problem into an equivalent one, with the goal of reducing the solving time. Good examples are the addition of new constraints, e.g., symmetry-breaking (cf., e.g., [28,8,12,22]), or implied constraints (cf., e.g., [30]), in order to, respectively, reduce the size of the search space, and to be able to perform better On the other hand, current systems for CP exhibit a neat separation between problem specifications and instances, usually adopting a two-level architecture for finding solutions: the specification is instantiated (or grounded) against the instance, and then an appropriate solver is invoked (cf. Figure 1).…”

Section: Introductionmentioning

confidence: 99%

Using a Theorem Prover for Reasoning on Constraint Problems

Cadoli

Mancini

2007

Applied Artificial Intelligence

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Abstract. Specifications of constraint problems can be considered logical formulae. As a consequence, it is possible to infer their properties by means of automated reasoning tools, with the goal of automatically synthesizing transformations that can make the solving process more efficient. The purpose of this paper is to link two important technologies: automated theorem proving (ATP) and constraint programming (CP). We report the results on using ATP technology for checking existence of symmetries, checking whether a given formula breaks a symmetry, and checking existence of functional dependencies in a specification. The output of the reasoning phase is a transformed constraint program, consisting in a reformulated specification and, possibly, a search strategy. We show our techniques on problems such as Graph coloring, Sailco inventory, and Protein folding.

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On the satisfiability of symmetrical constrained satisfaction problems

Cited by 99 publications

References 5 publications

Dominance rules in combinatorial optimization problems

Dominance rules in combinatorial optimization problems

SymChaff: exploiting symmetry in a structure-aware satisfiability solver

Using a Theorem Prover for Reasoning on Constraint Problems

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