Class Of Constraint Satisfaction Problems Research Articles

A note on some collapse results of valued constraints

Valued constraint satisfaction problem (VCSP) is an optimisation framework originally coming from Artificial Intelligence and generalising the classical constraint satisfaction problem (CSP). The VCSP is powerful enough to describe many important classes of problems. In order to investigate the complexity and expressive power of valued constraints, a number of algebraic tools have been developed in the literature. In this note we present alternative proofs of some known results without using the algebraic approach, but by representing valued constraints explicitly by combinations of other valued constraints.

The Complexity of Weighted Boolean #CSP

This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterized by a finite set $\mathcal{F}$ of nonnegative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that computing the partition function, i.e., the sum of the weights of all configurations, is $\text{{\sf FP}}^{\text{{\sf#P}}}$-complete unless either (1) every function in $\mathcal{F}$ is of “product type,” or (2) every function in $\mathcal{F}$ is “pure affine.” In the remaining cases, computing the partition function is in P.

Colouring, constraint satisfaction, and complexity

Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity — the k -colouring problem is polynomial-time solvable when k ≤ 2 , and NP-complete when k ≥ 3 . Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction. However, for general constraint satisfaction problems, dichotomy has only been conjectured. Although the conjecture remains unproven to this day, it has been driving much of the theoretical research on constraint satisfaction problems, which combines methods of logic, universal algebra, analysis, and combinatorics. Currently, this is a very active area of research, and it is our goal here to present some of the recent developments, updating some of the information in existing books and surveys, while focusing on both the mathematical and the computational aspects of the theory. Given the level of activity, we are only able to survey a fraction of the new work, with emphasis on our own areas of interest.

A UNIFYING FRAMEWORK FOR GENERALIZED CONSTRAINT ACQUISITION

When a practical problem can be modeled as a constraint satisfaction problem (CSP), which is a set of constraints that need to be satisfied, it can be solved using many constraint programming techniques. In many practical applications, while users can recognize examples of where a CSP should be satisfied or violated, they cannot articulate the specification of the CSP itself. In these situations, it can be helpful if the computer can take an active role in learning the CSP from examples of its solutions and non-solutions. This is called constraint acquisition. This paper introduces a framework for constraint acquisition in which one can uniformly define and formulate constraint acquisition problems of different types as optimization problems. The difference between constraint acquisition problems within the framework is not only in the type of constraints that need to be acquired but also in the learning objective. The generic framework can be instantiated to obtain specific formulations for acquiring classical, fuzzy, weighted or probabilistic constraints. The paper shows as an example how recent techniques for acquiring classical constraints can be directly obtained from the framework. Specifically, the formulation obtained from the framework to acquire classical CSPs with the minimum number of violated examples is equivalent to a simple pseudo-boolean optimization problem, thus being efficiently solvable by using many available optimization tools. The paper also reports empirical results on constraint acquisition methods to show the utility of the framework.

Dynamic structural symmetry breaking for constraint satisfaction problems

In recent years, symmetry breaking for constraint satisfaction problems (CSPs) has attracted considerable attention. Various general schemes have been proposed to eliminate symmetries. In general, these schemes may take exponential space or time to eliminate all the symmetries. We identify several classes of CSPs that encompass many practical problems and for which symmetry breaking for various forms of value or variable interchangeability is tractable using dedicated search procedures. We also show the limits of efficient symmetry breaking for such dominance-detection schemes by proving intractability results for some classes of CSPs.

Constraint Satisfaction, Logic and Forbidden Patterns

In the 1990s, Feder and Vardi attempted to find a large subclass of NP which exhibits a dichotomy, that is, where every problem in the subclass is either solvable in polynomial‐time or NP‐complete. Their studies resulted in a candidate class of problems, namely, those definable in the logic MMSNP. While it remains open as to whether MMSNP exhibits a dichotomy, for various reasons it remains a strong candidate. Feder and Vardi added to the significance of MMSNP by proving that, although MMSNP strictly contains CSP, the class of constraint satisfaction problems, MMSNP and CSP are computationally equivalent. We introduce here a new class of combinatorial problems, the class of forbidden patterns problems FPP, and characterize MMSNP as the finite unions of problems from FPP. We use our characterization to detail exactly those problems that are in MMSNP but not in CSP. Furthermore, given a problem in MMSNP, we are able to decide whether the problem is in CSP or not (this whole process is effective). If the problem is in CSP, then we can construct a template for this problem; otherwise, for any given candidate for the role of template, we can build a counterexample (again, this process is effective).

A multiagent evolutionary algorithm for constraint satisfaction problems

With the intrinsic properties of constraint satisfaction problems (CSPs) in mind, we divide CSPs into two types, namely, permutation CSPs and nonpermutation CSPs. According to their characteristics, several behaviors are designed for agents by making use of the ability of agents to sense and act on the environment. These behaviors are controlled by means of evolution, so that the multiagent evolutionary algorithm for constraint satisfaction problems (MAEA-CSPs) results. To overcome the disadvantages of the general encoding methods, the minimum conflict encoding is also proposed. Theoretical analyzes show that MAEA-CSPs has a linear space complexity and converges to the global optimum. The first part of the experiments uses 250 benchmark binary CSPs and 79 graph coloring problems from the DIMACS challenge to test the performance of MAEA-CSPs for nonpermutation CSPs. MAEA-CSPs is compared with six well-defined algorithms and the effect of the parameters is analyzed systematically. The second part of the experiments uses a classical CSP, n-queen problems, and a more practical case, job-shop scheduling problems (JSPs), to test the performance of MAEA-CSPs for permutation CSPs. The scalability of MAEA-CSPs along n for n-queen problems is studied with great care. The results show that MAEA-CSPs achieves good performance when n increases from 10(4) to 10(7), and has a linear time complexity. Even for 10(7)-queen problems, MAEA-CSPs finds the solutions by only 150 seconds. For JSPs, 59 benchmark problems are used, and good performance is also obtained.

A Simple Algorithm for Mal'tsev Constraints

A Mal'tsev operation is a ternary operation $\varphi$ that satisfies the identities $\varphi(x,y,y) = \varphi(y,y,x) = x$. Constraint satisfaction problems involving constraints invariant under a Mal'tsev operation constitute an important class of constraint satisfaction problems, which includes the affine satisfiability problem, subgroupand near subgroup constraints, and many others. It is also known that any tractable case of the counting constraint satisfaction problem involves only Mal'tsev constraints. The first algorithm solving the arbitrary constraint satisfaction problem with Mal'tsev constraints has been given by Bulatov. However, this algorithm is very sophisticated and relies heavily on advanced algebraic machinery. In this paper, we give a different and much simpler algorithm for this type of constraint.

A new tractable class of constraint satisfaction problems

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.

An efficient algorithm for a class of constraint satisfaction problems

We define the class of the so-called monotone constraint satisfaction problems (MON-CSP). MON-CSP forms a subclass of the class of min-closed (respectively, max-closed) constraint satisfaction problems of Jeavons and Cooper (Artificial Intelligence 79 (1995) 327). We prove that for all problems in the class MON-CSP there exists a very fast and very simple algorithm for testing feasibility. We then show that a number of well-known results from the literature are special cases of MON-CSP: (1) Satisfiability of Horn formulae; (2) graph homomorphisms to directed graphs with an X ̄ -numbering; (3) monotone integer programming with two variables per inequality; (4) project scheduling under AND/OR precedence constraints. Our results provide a unified algorithmic approach to all these problems.

Dynamic algorithms for classes of constraint satisfaction problems

Many fundamental tasks in artificial intelligence and in combinatorial optimization can be formulated as a Constraint Satisfaction Problem (CSP). It is the problem of finding an assignment of values for a set of variables, each defined on a finite domain of feasible values, subject to a given collection of constraints. Each constraint is defined over a set of variables and specifies the allowed combinations of values as a collection of tuples. In general, the problem of finding a solution to a CSP is NP-complete, but in some cases it has shown to be polynomially solvable. We consider the dynamic version of some polynomially solvable constraint satisfaction problems, and present solutions that are better than recomputing everything from scratch after each update. The updates we consider are either restrictions, i.e., deletions of values from existing constraints and introduction of new constraints, or relaxations, i.e., insertions of values or deletions of constraints.

A comparison of structural CSP decomposition methods

We compare tractable classes of constraint satisfaction problems (CSPs). We first give a uniform presentation of the major structural CSP decomposition methods. We then introduce a new class of tractable CSPs based on the concept of hypertree decomposition recently developed in Database Theory, and analyze the cost of solving CSPs having bounded hypertree-width. We provide a framework for comparing parametric decomposition-based methods according to tractability criteria and compare the most relevant methods. We show that the method of hypertree decomposition dominates the others in the case of general CSPs (i.e., CSPs of unbounded arity). We also make comparisons for the restricted case of binary CSPs. Finally, we consider the application of decomposition methods to the dual graph of a hypergraph. In fact, this technique is often used to exploit binary decomposition methods for nonbinary CSPs. However, even in this case, the hypertree-decomposition method turns out to be the most general method.

Hard, flexible and dynamic constraint satisfaction

Constraint satisfaction is a fundamental artificial intelligence technique offering a simple yet powerful representation. An increasing amount of attention has recently been paid to the development of constraint satisfaction techniques, and it has become clear that the original formulation of a static Constraint Satisfaction Problem (CSP) with hard, imperative constraints is insufficient to model many real problems. Two important extensions to the classical CSP framework which address some of these deficiencies are flexible and dynamic constraint satisfaction. This paper examines in detail classical, flexible and dynamic CSP. It reviews the motivations behind both extensions, and describes the techniques used to solve each type of problem. The paper employs a running example throughout to illustrate the ideas presented.

On Considering Constraints of Different Importance in Fuzzy Constraint Satisfaction Problems

Several real-world applications (e.g., scheduling, configuration, …) can be formulated as Constraint Satisfaction Problems (CSP). In these cases, a set of variables have to be settled to a value with the requirement that they satisfy a set of constraints. Classical CSPs are defined only by means of crisp (Boolean) constraints. However, as sometimes Boolean constraints are too strict in relation to human reasoning, fuzzy constraints were introduced. When fuzzy constraints are considered, human reasoning usually performs some compensation between alternatives. Thus other operators than t-norms are advisable. Besides of that, not all constraints can be considered with equal importance. In this paper we show that the WOWA operator can consider both aspects: compensation between constraints and constraints of different importance.

Learning solution preferences in constraint problems

. Usually, not all the solutions of a finite domain constraint satisfaction problem (CSP) are equally desirable: some of them may be preferred to others. However, classical CSPs do not allow for this more informative kind of knowledge representation. On the other hand, semiring-based CSPs (SCSPs), where a value is associated with each tuple in each constraint, generate solutions with a corresponding value attached that can be interpreted as the level of preference of that solution. Sometimes, however, even standard SCSPs are not enough, since one may know preferences over some of the solutions but have no idea on how to code this knowledge into the SCSP. In this paper we consider this situationand propose to address it by first defining a classical CSP and giving some examples of solution preferences, and then learning the corresponding SCSP that behaves as the initial CSP (that is, it has the same solutions) and matches the preferences specified in the examples. In other words, we use the examples as the training set, and we employ a learning scheme to adjust the values to be attached to the constraint tuples, such that the resulting solution preferences coincide with the examples. In this way, we make the SCSP framework more flexible, since it can be used also when it is difficult to assign values to tuples and instead it is easier to rate some of the solutions.

Constraint satisfaction problem with bilevel constraint: application to interpretation of over-segmented images

In classical finite-domain constraint satisfaction problems, the assumption made is that only one value is associated with only one variable. For example, in pattern recognition one variable is associated with only one segmented region. However, in practice, regions are often oversegmented which results in failure of any one to one mapping. This paper proposes a definition of finite-domain constraint satisfaction problems with bilevel constraints in order to take into account a many to one relation between the values and the variables. The additional level of constraint concerns the data assigned to the same complex variable. Then, we give a definition of the arc-consistency problem for bilevel constraint satisfaction checking. A new algorithm for arc consistency to deal with these problems is presented as well. This extension of the arc-consistency algorithm retains its good properties and has a time complexity in O( en 3 d 2) in the worst case. This algorithm was tested on medical images. These tests demonstrate its reliability in correctly identifying the segmented regions even when the image is over-segmented.

Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty

In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instantiations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instantiations are provided owing to the relaxation of some constraints. Fuzzy constraints are thus flexible. CSP notions of consistency and k-consistency can be extended to this framework and the classical algorithms used in CSP resolution (e.g., tree search and filtering) can be adapted without losing much of their efficiency. Most classical theoretical results remain applicable to FCSPs. In the paper, various types of constraints are modelled in the same framework. The handling of uncertain parameters is carried out in the same setting because possibility theory can account for both preference and uncertainty. The presence of uncertain parameters leads to ill-defined CSPs, where the set of constraints which defines the problem is not precisely known.

Consistency techniques for continuous constraints

We consider constraint satisfaction problems with variables in continuous, numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constraints. In particular, we show how globally consistent (also called decomposable) labelings of a constraint satisfaction problem can be computed. Our approach is based on approximating regions of feasible solutions by 2 k -trees, a representation commonly used in computer vision and image processing. We give simple and stable algorithms for computing labelings with arbitrary degrees of consistency. The algorithms can process constraints and solution spaces of arbitrary complexity, but with a fixed maximal resolution. Previous work has shown that when constraints are convex and binary, path-consistency is sufficient to ensure global consistency. We show that for continuous domains, this result can be generalized to ternary and in fact arbitrary n-ary constraints using the concept of (3,2)-relational consistency. This leads to polynomial-time algorithms for computing globally consistent labelings for a large class of constraint satisfaction problems with continuous variables.

Constraint logic programming for qualitative and quantitative constraint satisfaction problems

AI and OR approaches have complementary strengths: AI in domain-specific knowledge representation and OR in efficient mathematical computation. Constraint Logic Programming (CLP), which combines these complementary strengths of the AI and OR approach, is introduced as a new tool to formalize a special class of constraint satisfaction problems that include both qualitative and quantitative constraints. The CLP approach is contrasted with the Mixed Integer Programming (MIP) method from a model-theoretic view. Three relative advantages of CLP over MIP are analyzed: (1) representational economies for domain-specific heuristics, (2) partial solutions, and (3) ease of model revision. A case example of constraint satisfaction problems is implemented by MIP and CLP for comparison of the two approaches. The results exhibit those relative advantages of CLP with computational efficiency comparable to MIP.

A FRAMEWORK FOR INTEGRATING ARTIFICIAL NEURAL NETWORKS AND LOGIC PROGRAMMING

Many real-life problems belong to the class of constraint satisfaction problems (CSP’s), which are NP-complete, and some NP-hard, in general. When the problem size grows, it becomes difficult to program solutions and to execute the solution in a timely manner. In this paper, we present a general framework for integrating artificial neural networks and logic programming so as to provide an efficient and yet expressive programming environment for solving CSP’s. To realize this framework, we propose PROCLANN, a novel constraint logic programming language. The PROCLANN language retains the simple and elegant declarative semantics of constraint logic programming. Operationally, PROCLANN uses the standard goal reduction strategy in the frontend to generate constraints, and an efficient backend constraint-solver based on artificial neural networks. Its operational semantics is probabilistic in nature. We show that PROCLANN is sound and weakly complete. A novelty of PROCLANN is that while it is a committed-choice language, PROCLANN supports non-determinism, allowing the generation of multiple answers to a query. An initial prototype implementation of PROCLANN is constructed and demonstrates empirically that PROCLANN out-performs the state of art in constraint logic programming implementation on certain hard instances of CSP’s.