Valued Constraint Satisfaction Problems: Hard and Easy Problems (original) (raw)

Tractability in constraint satisfaction problems: a survey

Constraints, 2015

Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. 1. automatic recognition and resolution of easy instances within general-purpose solvers, supported by ANR Project ANR-10-BLAN-0210 and EPSRC grant EP/L021226/1.

High-Order Consistency in Valued Constraint Satisfaction

Constraints, 2005

k-consistency operations in constraint satisfaction problems (CSPs) render constraints more explicit by solving size-k subproblems and projecting the information thus obtained down to low-order constraints. We generalise this notion of k-consistency to valued constraint satisfaction problems (VCSPs) and show that it can be established in polynomial time when penalties lie in a discrete valuation structure. A generic definition of consistency is given which can be tailored to particular applications. As an example, a version of high-order consistency (face consistency) is presented which can be established in low-order polynomial time given certain restrictions on the valuation structure and the form of the constraint graph.

A filtering technique to achieve 2-consistency in constraint satisfaction problems

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.

Constraint satisfaction---a survey

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.

A Decomposition Technique for Max-CSP

European Conference on Artificial Intelligence, 2008

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 intro- duced by Freuder and Hubbe, we show that it is possible to exploit the arc-inconsistency counts, associated with each value of a net-

Modelling and Reformulating Constraint Satisfaction Problems

Constraint Programming (CP) is a powerful technology to solve combinatorial problems which are ubiquitous in academia and industry. The last ten years or so have witnessed significant research devoted to modelling and solving problems with constraints. CP is now a mature field and has been successfully used for tackling a wide range of real-life complex applications. However, such a technology is currently accessible to only a small number of experts.

Algorithms for Constraint Satisfaction Problems (CSPs

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.

An algebraic approach to constraint satisfaction problems

A constraint satisfaction problem, or CSP, can be reformulated as an integer linear programming problem. The reformulated problem can be solved via polynomial multiplication. If the CSP has n variables whose domain size is m, and if the equivalent programming problem involves M equations, then the number of solutions can be determined in time O(nm2 M −n ). This surprising link between search problems and algebraic techniques allows us to show improved bounds for several constraint satisfaction problems, including new simply exponential bounds for determining the number of solutions to the n-queens problem. We also address the problem of minimizing M for a particular CSP.

A New Approach for Weighted Constraint Satisfaction

Constraints, 2002

Abstract. We consider the Weighted Constraint Satisfaction Problem which is an important problem in Artificial Intelligence. Given a set of variables, their domains and a set of constraints between variables, our goal is to obtain an assignment of the variables to domain values such that ...