Optimization-based heuristics for maximal constraint satisfaction

Many problems in AI can be modeled as constraint satisfaction problems (CSPs). Hence the development of e ective solution techniques for CSPs is an important research problem. Forward checking (FC) with some other heuristics has been traditionally considered to be the best algorithm for solving CSPs while recently there have been a number of claims that maintaining arc consistency (MAC) is more e cient on large and hard CSPs. In this thesis, we p r o vide a systematic comparison empirically of the performances of the MAC a n d F C algorithms on large and hard CSPs. In particular, we compare their performance with regard to the size, constraint density and constraint t i g h tness of the problems. Though there is a trend that MAC e v entually outperforms FC on hard problems as we increase the problem size, we found that the superiority o f M A C o ver FC w ould not be revealed on the hard problems with low constraint t i g h tness and high constraint density u n til the size of these problems is quite large. We also devised a new FC algorithm | FC4, which s h o ws good performance on the hard problems with low constraint tightness and high constraint density. iv I w ould also like to thank Jean-Charles Regin of ILOG for providing his programs, and his assistance in my understanding his algorithms.

The objective of the Maximal Constraint Satisfaction Problem (Max-CSP) is to find an instantiation which minimizes the number of constraint violations in a constraint network. In this paper, inspired from the concept of inferred disjunctive constraints introduced by Freuder and Hubbe, we show that it is possible to exploit the arc-inconsistency counts, associated with each value of a network, in order to avoid exploring useless portions of the search space. The principle is to reason from the distance between the two best values in the domain of a variable, according to such counts. From this reasoning, we can build a decomposition technique which can be used throughout search in order to decompose the current problem into easier sub-problems. Interestingly, this approach does not depend on the structure of the constraint graph, as it is usually proposed. Alternatively, we can dynamically post hard constraints that can be used locally to prune the search space. The practical interest of our approach is illustrated, using this alternative, with an experimentation based on a classical branch and bound algorithm, namely PFC-MRDAC.

1996

Two methods are described for enhancing performance of branch and bound methods for overconstrained CSPs. These methods improve either the upper or lower bound, respectively, during search, so the two can be combined. Upper bounds are improved by using heuristic repair methods before search to find a good solution quickly, whose cost is used as the initial upper bound. The method for improving lower bounds is an extension of directed arc consistency preprocessing, used in conjunction with forward checking. After computing directed arc consistency counts, inferred counts are computed for all values based on minimum counts for values of adjacent variables that are later in the search order. This inference process can be iterated, so that counts are cascaded from the end to the beginning of the search order, to augment the initial counts. Improvements in time and effort are demonstrated for both techniques using random problems.

Artificial Intelligence Review, 2001

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, we present the problems that have been selected for the first international competition of CSP solvers. First, we introduce a succinct description of each problem and then, we present the two formats that have been used to represent the CSP instances.

Theoretical computer …, 2001

Many fundamental tasks in arti cial intelligence and in combinatorial optimization can be formulated as a Constraint Satisfaction Problem (CSP) 19]. The problem consists in nding an assignment of values for a set of n variables, each de ned on a nite domain of feasible values of size at most k, subject to a given collection of constraints. Each constraint is de ned over a set of variables and speci es the set of allowed combinations of values as a collection of tuples. In general the problem of nding a solution to a CSP is NP-complete, even if restricted to binary constraints. As an example, the graph coloring problem 11] can be formulated as a binary CSP, where each edge in the graph is associated to a constraint consisting of the collection of C 2 ?C pairs of allowed di erent colorings of the two endpoints with C colors: the resulting CSP is solvable if and only if the graph is colorable with C colors.

1998

Constraint satisfaction has been used as a term to cover a wide range of methods to solve problems stated in the form of a set of constraints. As the general constraint satisfaction problem (CSP) is NP-complete, initially the research focused on developing new and more efficient solution methods, resulting in an arsenal of algorithms. Recently, much attention has been paid on how to finetune the use of this arsenal, and to be able to judge which methods are promising for a given problem or problem-type. In the last few years different generalisations of the classical CSP have got much attention too, allowing to model a wider range of everyday problems. In this survey we introduce the classical CSP and the basic solution techniques as well as the ongoing research on the applicability of these methods and on extensions of the classical framework. After giving some introductory examples we define the most essential technical notions in order to explain different solution methods. First, we discuss constraint propagation algorithms, which transform the initially given CSP step by step to an equivalent, but smaller problem. Then we will introduce a family of constructive search algorithms, followed by methods exploiting the structure of the problem. Finally, we discuss the local and stochastic methods, also applicable to solve non-standard problems. The discussion of solution methods will be closed by addressing the issue of choosing a good algorithm for a given problem. Many practical applications have essential characteristics which do not "fit into" the classical formalism of CSP. The extension of the problem definition and appropriate solution methods will be dealt with in the final chapter.

Typical real world tasks such as configuration or design exhibit dynamic aspects which require extending the basic constraint satisfaction framework. In this paper we use the notion of conditional constraint satisfaction (CondCSP) as defined by [ Mittal and Falkenhainer, 1990 ] to provide a framework in which these problems can be formulated and solved. It has been shown in [ Soininen et al., 1999 ] that the decision problem for CondCSP is NP -complete similar to standard CSPs. Solving standard CSPs applies various degrees of local consistency to avoid recomputing infeasible branches in the search tree. In a similar manner, we define new types of local consistency aimed at solving conditional CSPs more e#ciently. The di#erent algorithms are compared using a series of randomly generated problems.

2007

It might be said that there are five basic tree search algorithms for the constraint satisfaction problem (csp), namely, naive backtracking (BT), backjumping (BJ), conflict-directed backjumping (CBJ), backmarking (BM), and forward checking (FC). In broad terms, BT, BJ, and CBJ describe different styles of backward move (backtracking), whereas BT, BM, and FC describe different styles of forward move (labeling of variables). This paper presents an approach that allows base algorithms to be combined, giving us new hybrids.

Este é um espaço que considero muito importante em meus livros. É o momento de reconhecimento ao apoio direto ou indireto de pessoas importantes para mim, durante o processo de pesquisa e escrita do livro. Se você já leu outros livros meus, certamente achará esta introdução parecida, mas aqui, nesta seção, sou assim. Não posso jamais deixar de agradecer ao apoio de meus filhos, pessoas chaves em minha vida. No momento em que me encontro, pessoal e profissionalmente, duas pessoinhas são muito importantes para os momentos de relaxamento: Maria Clara e Vicente Dirceu. Meus caçulas, minhas joias mais preciosas. Meu filho Gabriel, conversamos quase todos os dias, na maioria das vezes você me dando conselhos. Obrigado, meus amados filhos. Um hiato pessoal, me permitam. Conheci a dança este ano. Vocês não sabem como esta atividade faz bem para a vida, para a mente e o poder que tem em nos trazer felicidade. Se não sabem dançar, aprendam. Se sabem dançar, dancem mais. Bolero, tango, samba, forró, ou o gênero de que você gostar. Na dança, em meu caso, todas as pessoas e grupos que conheci são maravilhosos, do bem, todos dispostos a ajudar e ensinar. Momentos espetaculares são vividos e perpetuados com esta atividade e, para o desenvolvimento, nos pontos de dificuldades, relaxa a mente e as soluções chegam. Dance! Um agradecimento importante ao Davi Marcondes Rocha, exaluno, ex-colega de doutorado, e sempre amigo. Hoje colega de AGRADECIMENTOS E-book gerado especialmente para Lucas Romagnoli