Solving constraint satisfaction problems with heuristic-based evolutionary algorithms

This paper introduces a genetic local search algorithm for bi­ nary constraint satisfaction problems. The core of the algo­ rithm consists of an ad-hoc optimization procedure followed by the application of blind genetic operators. A standard set of benchmark instances is used in order to assess the performance of the algorithm. The results indicate that this apparently naive hybridation of a genetic algorithm with lo­ cal search yields a rather powerful heuristic algorithm for random binary constraint satisfaction problems.

The order in which the variables are assigned can have an enormous impact on the time taken by a backtracking search algorithm to solve a constraint satisfaction problem (CSP). The Brelaz heuristic is a dynamic variable ordering heuristic which has been shown to give good results for some classes of binary CSPs when the constraint graph is not complete. Its advantage over the simpler smallest-domain heuristic is that it uses information about the constraint graph. This paper uses theoretical work by Nudel to assess the performance of the Brelaz heuristic. Nudel’s work gives the expected number of nodes at each level of the search tree when using the forward checking algorithm to find all solutions to a CSP, given a specified order of the variables. From this, optimal static orderings are found for a sample of small binary CSPs. The optimal orderings are used to learn rules for a static ordering heuristic, which are converted into modifications to the Brelaz heuristic. The improved heuristic is shown to halve the mean search cost of solving sparse random binary CSPs with 50 variables, at the phase transition. However, our modifications, and the Brelaz heuristic itself, are mainly in the form of improved tie-breakers for the smallest-domain heuristic, which the results suggest is still the basis of good heuristics for this class of problem.

Constructing an asymptotic phase transition in random binary constraint satisfaction problems

A recent theoretical result by Achlioptas et al. shows that many models of random binary constraint satisfaction problems become trivially insoluble as problem size increases. This insolubility is partly due to the presence of ‘flawed variables,’ variables whose values are all ‘flawed’ (or unsupported). In this paper, we analyse how seriously existing work has been affected. We survey the literature to identify experimental studies that use models and parameters that may have been affected by flaws. We then estimate theoretically and measure experimentally the size at which flawed variables can be expected to occur. To eliminate flawed values and variables in the models currently used, we introduce a ‘flawless’ generator which puts a limited amount of structure into the conflict matrix. We prove that such flawless problems are not trivially insoluble for constraint tightnesses up to 1/2. We also prove that the standard models B and C do not suffer from flaws when the constraint tightness is less than the reciprocal of domain size. We consider introducing types of structure into the constraint graph which are rare in random graphs and present experimental results with such structured graphs.

A new branch-and-filter exact algorithm for binary constraint satisfaction problems

Constraint satisfaction arises in many domains in different forms. Search and inference compete for solving constraint satisfaction problems (CSPs) but the most successful approaches are those which benefit from both techniques. Based on this idea, this article introduces a new scheme for solving the general Max-CSP problem. The new approach exploits the simplicity and efficiency of a modified Particle Swarm Optimization and the advantage of adaptable inference levels offered by the Mini-Bucket Elimination algorithm. Experiments conducted on binary CSPs using different levels of inference are illustrative for the inference/search trade-off. Comparative studies highlight the differences between our stochastic population-based method and the systematic search performed by a Branch and Bound algorithm.

We investigate rules which allow variable elimination in binary CSP (constraint satisfaction problem) instances while conserving satisfiability. We study variable-elimination rules based on the language of forbidden patterns enriched with counting and quantification over variables and values. We propose new rules and compare them, both theoretically and experimentally. We give optimised algorithms to apply these rules and show that each define a novel tractable class. Using our variable-elimination rules in preprocessing allowed us to solve more benchmark problems than without.

We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the (d,k,t)‐model, and binary constraint satisfaction problems with domain size three. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivatedthe multitude of works that have been devoted to developing techniques that simplify CSP instances before or duringtheir resolution.The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of valuedomains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrizedgeneralizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhoodsubstitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination,which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs byidentifying a new tractable class of binary CSP.

The aim of this paper is to establish a connection between the propositional logic and the constraint based reasoning frameworks. This work is based on a translation of the satisfiability problem (SAT) into the binary constraint-satisfaction problem (CSP). The structure of the SAT problem and its associated CSP are then exploited together for characterizing tractable SAT problems, increasing the effectiveness of the classical reduction rules: unit clause and monotone literal rules, and expressing the arc and path consistency concepts with logical inference rules. This study leads to compare the behaviors of the DP and MAC procedures for solving respectively a SAT instance and its binary CSP expression.

This paper provides a technique for solving general constraint satisfaction problems (CSPs) with continuous variables. Constraints are represented by a hierarchical binary decomposition of the space of feasible values. We propose algorithms for path- and higher degrees of consistency based on logical operations defined on this representation and demonstrate that the algorithms terminate in polynomial time. We show that, in analogy to convex temporal problems and discrete row-convex problems, convexity properties of the solution spaces can be exploited to compute minimal and decomposable networks using path consistency algorithms. Based on these properties, we also show that a certain class of non binary CSPs can be solved using strong 5-consistency.

Conventional techniques for the constraint satisfaction problem (CSP) have had considerable success in their applications. However, there are many areas in which the performance of the basic approaches may be improved. These include heuristic ordering of certain tasks performed by the CSP solver, hybrids which combine compatible solution techniques and graph based methods which exploit the structure of the constraint graph representation of a CSP. Also, conventional constraint satisfaction techniques only address problems with hard constraints (i.e. each of which are completely satisfied or completely violated, and all of which must be satisfied by a valid solution). Many real applications require a more flexible approach which relaxes somewhat these rigid requirements. To address these issues various approaches have been developed. This paper attempts a systematic review of them.

In this paper, Coevolutionary Genetic Algorithm for solving Constraint Satisfaction Problems (CSPs) is proposed. It consists of two Genetic Algorithms (GAs): a traditional GA and another GA to search for good schemata in the former GA. These GAs evolve in two levels, i.e., phenotype-level and schema-level, and affect with each other through genetic operations. To search for solutions effectively, we devise new genetic operator by utilizing search mechanism of solution synthesis approach used in CSP community. Computational results on general CSPs confirm the effectiveness of our approach.

Proceeding of: IEEE Congress on Evolutionary Computation (CEC 2009), May 18-21 (Monday - Thursday), 2009, Trondheim, Norway.

Generalized arc-consistency propagation is predominantly used in constraint solvers to efficiently prune the search space when solving constraint satisfaction problems. Although many practical applications can be modelled as distributed constraint satisfaction problems, no distributed arc-consistency algorithms so far have considered the privacy of individual agents. In this paper, we propose a new distributed arc-consistency algorithm, called $$\mathsf {DisAC3.1}$$ , which leaks less private information of agents than existing distributed arc-consistency algorithms. In particular, $$\mathsf {DisAC3.1}$$ uses a novel termination determination mechanism, which allows the agents to share domains, constraints and communication addresses only with relevant agents. We further extend $$\mathsf {DisAC3.1}$$ to $$\mathsf {DisGAC3.1}$$ , which is the first distributed algorithm that enforces generalized arc-consistency on k-ary ( $$k\ge 2$$ ) constraint satisfaction problems. Theoretical analyses show that our algorithms are efficient in both time and space. Experiments also demonstrate that $$\mathsf {DisAC3.1}$$ outperforms the state-of-the-art distributed arc-consistency algorithm and that $$\mathsf {DisGAC3.1}$$ ’s performance scales linearly in the number of agents.