Efficient structural symmetry breaking for constraint satisfaction problems

Symmetry breaking for constraint satisfaction problems (CSPs) has attracted considerable attention in recent years. Various general schemes have been proposed to eliminate symmetries. In general, these schemes may take exponential space or time to eliminate all the symmetries. We identify several classes of CSPs that encompass many practical problems and for which symmetry breaking for various forms of value and variable interchangeability is tractable using dedicated search procedures or symmetry-breaking constraints that allow nogoods and their symmetrically equivalent solutions to be stored and checked efficiently

Symmetry Breaking Constraints: Recent Results

Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful cases: symmetry breaking constraints for row and column symmetry, and symmetry breaking constraints for eliminating value symmetryComment: To appear in Proceedings of Twenty-Sixth Conference on Artificial Intelligence (AAAI-12

Improving the Computational Efficiency in Symmetrical Numeric Constraint Satisfaction Problems

Models are used in science and engineering for experimentation, analysis, diagnosis or design. In some cases, they can be considered as numeric constraint satisfaction problems (NCSP). Many models are symmetrical NCSP. The consideration of symmetries ensures that NCSP-solver will find solutions if they exist on a smaller search space. Our work proposes a strategy to perform it. We transform the symmetrical NCSP into a newNCSP by means of addition of symmetry-breaking constraints before the search begins. The specification of a library of possible symmetries for numeric constraints allows an easy choice of these new constraints. The summarized results of the studied cases show the suitability of the symmetry-breaking constraints to improve the solving process of certain types of symmetrical NCSP. Their possible speedup facilitates the application of modelling and solving larger and more realistic problems.Ministerio de Ciencia y Tecnología DIP2003-0666-02-

Biased landscapes for random Constraint Satisfaction Problems

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure

Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Ground-state configuration space heterogeneity of random finite-connectivity spin glasses and random constraint satisfaction problems

We demonstrate through two case studies, one on the p-spin interaction model and the other on the random K-satisfiability problem, that a heterogeneity transition occurs to the ground-state configuration space of a random finite-connectivity spin glass system at certain critical value of the constraint density. At the transition point, exponentially many configuration communities emerge from the ground-state configuration space, making the entropy density s(q) of configuration-pairs a non-concave function of configuration-pair overlap q. Each configuration community is a collection of relatively similar configurations and it forms a stable thermodynamic phase in the presence of a suitable external field. We calculate s(q) by the replica-symmetric and the first-step replica-symmetry-broken cavity methods, and show by simulations that the configuration space heterogeneity leads to dynamical heterogeneity of particle diffusion processes because of the entropic trapping effect of configuration communities. This work clarifies the fine structure of the ground-state configuration space of random spin glass models, it also sheds light on the glassy behavior of hard-sphere colloidal systems at relatively high particle volume fraction.Comment: 26 pages, 9 figures, submitted to Journal of Statistical Mechanic

Symmetries of Symmetry Breaking Constraints

Symmetry is an important feature of many constraint programs. We show that any symmetry acting on a set of symmetry breaking constraints can be used to break symmetry. Different symmetries pick out different solutions in each symmetry class. We use these observations in two methods for eliminating symmetry from a problem. These methods are designed to have many of the advantages of symmetry breaking methods that post static symmetry breaking constraint without some of the disadvantages. In particular, the two methods prune the search space using fast and efficient propagation of posted constraints, whilst reducing the conflict between symmetry breaking and branching heuristics. Experimental results show that the two methods perform well on some standard benchmarks.Comment: To appear in the Proceedings of the Ninth International Workshop on Symmetry and Constraint Satisfaction Problems, held alongside the 15th International Conference on Principles and Practice of Constraint Programming (CP 2009), Lisbon, Portuga

Glassy Behavior and Jamming of a Random Walk Process for Sequentially Satisfying a Constraint Satisfaction Formula

Random $K$-satisfiability ($K$-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random $K$-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process {\tt SEQSAT}, which satisfies a $K$-SAT formula in a sequential manner. Before satisfying each newly added clause, {\tt SEQSAT} walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density $\alpha$ of the satisfied subformula is less than certain value $\alpha_{cm}$; however it slows down considerably as $\alpha > \alpha_{cm}$ and finally reaches a jammed state at $\alpha \approx \alpha_{j}$. The glassy dynamical behavior of {\tt SEQSAT} for $\alpha \geq \alpha_{cm}$ probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point $\alpha_j$ is larger than the solution space clustering transition point $\alpha_d$, and its value can be predicted by a long-range frustration mean-field theory. For random $K$-SAT with $K\geq 4$, however, our simulation results indicate that $\alpha_j = \alpha_d$. The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.Comment: 10 pages, 6 figures, 1 table, a mistake of numerical simulation corrected, and new results adde