Nogood recording for valued constraint satisfaction problems

Constraint programming offers a declarative approach to solving problems modeled as constraint satisfaction problems (CSPs). However, the precise specification of a set of constraints is sometimes not available, but may have to be learned, for instance, from a set of examples of its solutions and non-solutions. In general, one may wish to learn generalized CSPs involving classical, fuzzy, weighted or probabilistic constraints, for example. This paper introduces a unifying framework for CSP learning. The framework is generic in that it can be instantiated to obtain specific formulations for learning classical, fuzzy, weighted or probabilistic CSPs. In particular, a new formulation for classical CSP learning, which minimizes the number of examples violated by candidate CSPs, is obtained by instantiating the framework. This formulation is equivalent to a simple pseudo-boolean optimization problem, thus being efficiently solvable using many optimization tools.

Nogood recording is a dynamic learning technique widely applied to solve CSP (constraint satisfaction problems). It is highly effective in reducing the search space for SAT (satisfiability) problems. While SAT is NP-complete, the problem restricted to binary clauses (2SAT) is solvable in linear time. We can improve SAT solving by incorporating 2SAT solving techniques. In this paper we investigate extending nogood recording to make use of binary clause resolution. Our experiments show that nogoods generated from binary resolution can significantly reduce the search space, and size of nogoods generated, as well as the search time.

Many AI synthesis problems such as planning, scheduling or design may be encoded in a constraint satisfaction problem (CSP). A CSP is typically defined as the problem of finding any consistent labeling for a fixed set of variables satisfying all given constraints between these variables. However, for many real tasks, the set of constraints to consider may evolve because of the environment or because of user interactions. The problem considered here is the solution maintenance problem in such a dynamic CSP (DCSP). The authors propose a new class of constraint recording algorithms called Nogood Recording that may be used for solving both static and dynamic CSPs. It offers an interesting compromise, polynomially bounded in space, between an ATMS-like approach and the usual static constraint satisfaction algorithms.

A note on some collapse results of valued constraints

It has been shown that evolutionary algorithms are able to construct suitable search strategies for classes of Constraint Satisfaction Problems (CSPs) in Constraint Programming. This paper is an explanation of the use of multi-objective optimisation in contrast to simple additive weighting techniques with a view to develop search strategies to classes of CSPs. A hierarchical scheme is employed to select a candidate strategy from the Pareto frontier for final evaluation. The results demonstrate that multi-objective optimisation significantly outperforms the single objective scheme in the same number of objective evaluations. In situations where strategies developed for a class of problems fail to extend to unseen problem instances of the same class, it is found that the structure of the underlying CSPs do not resemble those employed in the training process.

The N-Queen problem is a classical combinatorial optimization and constraint satisfaction problem that involves placing N queens on an N × N chessboard such that no two queens threaten each other. Due to its exponential search space and increasing computational complexity with larger board sizes, the problem serves as an important benchmark for evaluating the efficiency of heuristic and metaheuristic optimization techniques. This study presents a comprehensive performance analysis of the Simulated Annealing (SA) optimization approach for solving the N-Queen problem across varying board dimensions. The proposed methodology models the N-Queen problem as an optimization task by defining an objective function that minimizes the number of conflicting queen pairs. Simulated Annealing is employed to iteratively explore the solution space, allowing probabilistic acceptance of suboptimal solutions to escape local minima. Key parameters, including initial temperature, cooling schedule, and termination criteria, are systematically tuned to evaluate their impact on convergence speed and solution quality. Experimental results demonstrate that the SA-based approach effectively finds valid solutions for large values of N with high success rates and reduced computational effort compared to exhaustive and deterministic methods. Performance metrics such as solution accuracy, convergence time, number of iterations, and conflict reduction rate are analyzed and discussed. The results indicate that Simulated Annealing offers a robust and scalable solution for the N-Queen problem, particularly for higher-dimensional instances where traditional methods become impractical. Overall, this study highlights the suitability of Simulated Annealing as an efficient metaheuristic for solving complex combinatorial optimization problems and provides insights into its performance characteristics when applied to the N-Queen problem.

This is a summary of the PhD thesis defended by the author in September 2011 at the Ecole des Mines de Nantes (EMNantes). The thesis was advised by Narendra Jussien (EMNantes) and co-advised by Louis-Martin Rousseau (Ecole Polytechnique de Montreal—Poly), Christelle Gueret (EMNantes) and Andre Langevin (Poly). The manuscript is written in French and can be obtained from http://tel.archives-ouvertes. fr/tel-00630122/fr/ or by email from arnaud.malapert@unice.fr. The focus of the dissertation is to study the resolution of shop and batch scheduling problems derived from real-world applications using constraint programming. The remainder of this abstract summarizes the results of the dissertation. In shop problems, n jobs, consisting each of m tasks, must be processed on m machines. The processing orders of tasks which belong to a job can vary: global order (flow shop); order per job (job shop); no order (open shop). We consider the construction of non-preemptive schedules of minimal makespan, which is NP-Hard for m ≥ 3. The study and classification of models and search algorithms show that one of the major challenge of solving optimally these problems is to provide good solutions as quick as the metaheuristics. Therefore, we introduce a new algorithm characterized by its: simplicity, most of its components are available in constraint solvers; flexibility, two constraint models offer different trade-offs among propagation strength, speed and simplicity; genericity, it is applicable to any disjunctive scheduling problem. The algorithm relies on recent constraint filtering algorithm and (randomized) branching techniques with new upper bound heuristics. The algorithm starts with a randomized constructive heuristic (without propagation) which initializes the upper bound so that the selection and propagation of initial choices are improved. During the search, the filtering is based on the disjunctive graph model and disjunctive global constraints, whereas the branching is conducted by adding precedences to the disjunctive graph with the profile strategy. Besides, randomized restart policies combined with nogood recording allow to search diversification and learning from restarts. Indeed, nogoods

Practical implementation of random binning is one of the key challenges in achieving the largest available rate regions for many multiuser channels. This paper explores the use of low-density parity-check (LDPC) like codes for a particular kind of deterministic broadcast channel called the Blackwell channel and illustrates that random linear codes can be used to construct practical binning schemes at rates close to the capacity region of the Blackwell channel. The key ingredient is an encoding algorithm known as which is a generalization of the well-known belief propagation algorithm for LDPC codes. Survey propagation has been previously devised for a class of constraint satisfaction problems called K-SAT. This paper shows that the encoding problem for the Blackwell channel contains the same features as the constraint satisfaction problem and that the survey propagation algorithm, when concatenated with an outer error correcting code, works well at rates close to the Blackwell channel capacity region

Constraint Satisfaction with a Multi-Dimensional Domain

In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems containing all known polynomially solvable problems. We introduce a new class of problems called para-primal problems, incomparable with the families identified by Feder and Vardi (1998) and we prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal and homogeneous algebras.

The expressiveness of First Order Logic (FOL) languages is severely counterbalanced by the complexity of matching formulas on a universe. Matching is an instance of the class of Constraint Satisfaction Problems (CSP), which have shown to undergo a phase transition with respect to two order parameters: constraint density and constraint tightness. This paper analyzes the problem of satisfying FOL Horn clauses in the light of these recent results. By means of an extensive experimental analysis, we show how Horn clause verification exhibits a typical phase transition with respect to the number of binary (or greater arity) predicates, and with respect to the ratio between the number of constants in the universe and the cardinality of the basic predicates extension.

We introduce a generalization of the spreadsheet paradigm, called Knowledgesheet, for solving a class of constraint satisfaction problems. The traditional spreadsheet paradigm is based on attaching arithmetic expressions to individual cells and then evaluating them; our Knowledgesheet interface instead allows finite domain constraints to be attached to individual cells that are then solved to obtain a solution. This extension provides an easy-to-use interface for solving a large class of constraint satisfaction problems--those whose specification and solution conforms to a 2-dimensional structure, e.g., scheduling problems, timetabling problems, etc. A prototype for the Knowledgesheet has been developed and applied to solve many different types of problems.

Nogood recording is a form of learning that has been shown useful for solving constraint satisfaction problems. One simple approach involves recording nogoods that are extracted from the rightmost branches of the successive trees built by a backtrack search algorithm with restarts. In this paper, we propose several mechanisms to reason with so-called increasing-nogoods that exactly correspond to the states reached at the end of each search run. Interestingly, some similarities that can be observed between increasing-nogoods allow us to propose new original ways of dynamically combining them in order to improve the overall filtering capability of the learning system. Our preliminary results show the practical interest of our approach.

Local consistency algorithms, like arc consistency (AC) algorithms, are polynomial-time algorithms that prune the search space of constraint satisfaction problems (CSPs). In this paper, we present connections between message passing algorithms and AC for semiring-based CSPs (SCSPs) and valued CSPs (VCSPs), two well-established frameworks that generalize CSPs. Message passing algorithms are well known distributed search algorithms for solving many combinatorial problems in artificial intelligence, probabilistic reasoning, and information theory. However, the relationship between message passing algorithms and SCSPs or VCSPs still remains understudied. Towards this end, we propose the best-O message passing (BOMP) algorithm for SCSPs and VCSPs. We prove that, unlike other standard message passing algorithms which are in general not guaranteed to converge, the BOMP algorithm guarantees convergence for SCSPs and specific subclasses of VCSPs. We also theoretically study the relationship between the BOMP algorithm and AC on SCSPs, and empirically study the quality of the solutions produced by the BOMP algorithm for VCSPs.

In this paper we discuss the possibilities of applying genetic algorithms (GA) for solving constraint satisfaction problems (CSP). We point out how the greediness of deterministic classical CSP solving techniques can be counterbalanced by the random mechanisms of GAs. We tested our ideas by running experiments on four different CSPs: N-queens, graph 3-colouring, the traffic lights and the Zebra problem. Three of the problems have proven to be GA-easy, and even for the GA-hard one the performance of the GA could be boosted by techniques familiar in classical methods. Thus GAs are promising tools for solving CSPs. In the discussion, we address the issues of non-solvable CSPs and the generation of all the solutions.