Depth-First Search Techniques : A Short Survey Roman Barták

2012

Arc-Consistency algorithms are the most commonly used filtering techniques to prune the search space in Constraint Satisfaction Problems (CSPs). 2-consistency is a similar technique that guarantees that any instantiation of a value to a variable can be consistently extended to any second variable. Thus, 2-consistency can be stronger than arc-consistency in binary CSPs. In this work we present a new algorithm to achieve 2consistency called 2-C4. This algorithm is a reformulation of AC4 algorithm that is able to reduce unnecessary checking and prune more search space than AC4. The experimental results show that 2-C4 was able to prune more search space than arc-consistency algorithms in non-normalized instances. Furthermore, 2-C4 was more efficient than other 2-consistency algorithms presented in the literature.

A bit of history ... Constraints have recently emerged as a research area that combines researchers from a number of fields, including Artificial Intelligence, Programming Languages, Symbolic Computing and Computational Logic. Constraint networks and constraint satisfaction problems have been studied in Artificial Intelligence starting from the seventies (Montanary, 1974), (Waltz, 1975). Systematic use of constraints in programming has started in the eighties (Gallaire, 1985), (Jaffar, Lassez, 1987). The constraint satisfaction origins from Artificial Intelligence where the problems like scene labelling was studied (Waltz, 1975). The scene labelling problem is probably the first constraint satisfaction problem that was formalised. The goal is to recognise the objects in the scene by interpreting lines in the drawings. First, the lines or edges are labelled, i.e., they are categorised into few types, namely convex (+), concave (-) and occluding edges (<). In some advanced systems, the shadow border is recognised as well. There are a lot ways how to label the scene (exactly 3 n , where n is a number of edges) but only few of them has any 3D meaning. The idea how to solve this combinatorial problem is to find legal labels for junctions satisfying the constraint that the edge has the same label at both ends. This reduces the problem a lot because there are only a very limited number of legal labels for junctions. ... and some applications. Constraint programming has been successfully applied in numerous domains. Recent applications include computer graphics (to express geometric coherence in the case of scene analysis), natural language processing (construction of efficient parsers), database systems (to ensure and/or restore consistency of the data), operations research problems (like optimisation problems), molecular biology (DNA sequencing), business applications (option trading), electrical engineering (to locate faults), circuit design (to compute layouts), etc. Current research in this area deals with various foundational issues, with implementation aspects, and with new applications of constraint programming. What does the constraint programming deal with? Constraint programming is the study of computational systems based on constraints. The idea of constraint programming is to solve problems by stating constraints (conditions, properties, requirements) which must be satisfied by the solution.

2016

A perennial problem in hybrid backtrack CSP search is how much local consistency processing should be done to achieve the best efficiency. This can be divided into two separate questions: (1) how much work should be done before the actual search begins, i.e. during preprocessing?, and (2) how much of the same processing should be interleaved with search? At present there are two leading approaches to establishing stronger consistencies than the basic arc consistency maintenance that is done in most solvers. On the one hand there are various kinds singleton arc consistency that can be used; on the other there are several variants of restricted path consistency. To date these have not been compared directly. The present work attempts to do this for a variety of problems, and in so doing, it also provides an empirical evaluation of the preprocessing versus search processing issue. Comparisons are made using the domain/degree and domain/weighted degree variable ordering heuristics. In g...

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.

2000

In this paper, we present a novel discrepancy-based search technique implemented as an instance of the generic search procedures framework introduced in [10]. Our empirical results indicate that the Discrepancy-Bounded Depth First Search (DBDFS) procedure exhibits a number of good properties. As a discrepancy based search technique, it is able to quickly find solution with low deviation from the optimal solution. In addition, it revisits fewer nodes than other discrepancy-based techniques (e.g., Limited Discrepancy Search), making it a good candidate for proving optimal solutions.

Artificial Intelligence, 1980

In this paper we explore the number of tree search operations required to solve binary constraint satisfaction problems. We show analytically and experimentally that the two principles of first trying the places most likely to fail and remembering what has been done to avoid repeating the same mistake twice improve the standard backtracking search. We experimentally show that a lookahead procedure called forward checking (to anticipate the future) which employs the most likely to fail principle performs better than standard backtracking, Ullman's, Waltz's, Mackworth's, and Haralick's discrete relaxation in all cases tested, and better than Gaschnig's backmarking in the larger 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.

1994

The min-conflict heuristic (Minton et al. 1992) has been introduced into backtracking algorithms and iterative improvement algorithms as a powerful heuristic for solving constraint satisfaction problems. Backtracking algorithms become inefficient when a bad partial solution is constructed, since an exhaustive search is required for revising the bad decision. On the other hand, iterative improvement algorithms do not construct a consistent partial solution and can revise a bad decision without exhaustive search.

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.