Ants can solve constraint satisfaction problems

We describe a novel incomplete approach for solving constraint satisfaction problems (CSPs) based on the ant colony optimization (ACO) metaheuristic. The idea is to use artificial ants to keep track of promising areas of the search space by laying trails of pheromone. This pheromone information is used to guide the search, as a heuristic for choosing values to be assigned to variables. We first describe the basic ACO algorithm for solving CSPs and we show how it can be improved by combining it with local search techniques. Then, we introduce a preprocessing step, the goal of which is to favor a larger exploration of the search space at a lower cost, and we show that it allows ants to find better solutions faster. Finally, we evaluate our approach on random binary problems     

启发式回溯算法求解约束满足问题

针对一个典型的具有可变取值域的随机约束满足问题,提出了利用度启发式策略和最少约束值启发式策略来选择变量进行赋值的不完备回溯算法。该算法首先通过度启发式来确定待赋值变量的顺序,然后利用最少约束值启发式对选择的变量进行赋值,最后在有限时间内通过回溯得到变量的一组取值。用此算法对由RB模型生成的随机实例进行求解,实验结果表明,与经典的回溯算法相比,该算法具有显著的优越性。在控制参数(即约束紧度)进入相变区域时,该算法能在较短的时间内有效地找到实例的解。     

Applying constraint satisfaction approach to solve product configuration problems with cardinality-based configuration rules

In this paper, the product configuration problems that are characterized by cardinality-based configuration rules are dealt with. Novel configuration rules including FI and EI rules are presented to clarify the semantics of inclusion rules when cardinalities and hierarchies of products are encountered. Then, a configuration graph is proposed to visualize structural rules and configuration rules in product configuration problem. An encoding approach is elaborated to transform the configuration graph as a CSP (Constraint Satisfaction Problem). As a consequence, existing CSP solver, i.e. JCL (Java Constraint Library), is employed to implement the configuration system for product configuration problem with cardinality-related configuration rules. A case study of a bus configuration is used throughout this paper to illustrate the effectiveness of the presented approach.     

基于动态度的回溯算法求解大值域约束满足问题

针对一个具有精确可满足性相变现象的大值域随机约束满足问题,提出了两种启发式动态回溯算法,即基于动态度的ddeg-MAC（dynamic degree-maintaining arc consistency）回溯算法和基于值域与动态度比值的dom/ddeg-MAC（dom/dynamic degree-maintaining arc consistency）回溯算法。这两种算法分别基于ddeg和dom/ddeg挑选变量,利用维持弧相容（MAC）技术为挑选的变量进行赋值。当赋值无法进行时,再执行动态回溯修正变量的赋值。数值实验结果表明：在控制参数非常接近理论相变点时,算法仍然能够有效地找到问题的解。与经典回溯算法相比,这两种启发式动态回溯算法具有显著的优越性。     

Lexicographically-ordered constraint satisfaction problems

We describe a simple CSP formalism for handling multi-attribute preference problems with hard constraints, one that combines hard constraints and preferences so the two are easily distinguished conceptually and for purposes of problem solving. Preferences are represented as a lexicographic order over complete assignments based on variable importance and rankings of values in each domain. Feasibility constraints are treated in the usual manner. Since the preference representation is ordinal in character, these problems can be solved with algorithms that do not require evaluations to be represented explicitly. This includes ordinary CSP algorithms, although these cannot stop searching until all solutions have been checked, with the important exception of heuristics that follow the preference order (lexical variable and value ordering). We describe relations between lexicographic CSPs and more general soft constraint formalisms and show how a full lexicographic ordering can be expressed in the latter. We discuss relations with (T)CP-nets, highlighting the advantages of the present formulation, and we discuss the use of lexicographic ordering in multiobjective optimisation. We also consider strengths and limitations of this form of representation with respect to expressiveness and usability. We then show how the simple structure of lexicographic CSPs can support specialised algorithms: a branch and bound algorithm with an implicit cost function, and an iterative algorithm that obtains optimal values for successive variables in the importance ordering, both of which can be combined with appropriate variable ordering heuristics to improve performance. We show experimentally that with these procedures a variety of problems can be solved efficiently, including some for which the basic lexically ordered search is infeasible in practice.     

Solving quantified constraint satisfaction problems

We make a number of contributions to the study of the Quantified Constraint Satisfaction Problem (QCSP). The QCSP is an extension of the constraint satisfaction problem that can be used to model combinatorial problems containing contingency or uncertainty. It allows for universally quantified variables that can model uncertain actions and events, such as the unknown weather for a future party, or an opponent's next move in a game. In this paper we report significant contributions to two very different methods for solving QCSPs. The first approach is to implement special purpose algorithms for QCSPs; and the second is to encode QCSPs as Quantified Boolean Formulas and then use specialized QBF solvers. The discovery of particularly effective encodings influenced the design of more effective algorithms: by analyzing the properties of these encodings, we identify the features in QBF solvers responsible for their efficiency. This enables us to devise analogues of these features in QCSPs, and implement them in special purpose algorithms, yielding an effective special purpose solver, QCSP-Solve. Experiments show that this solver and a highly optimized QBF encoding are several orders of magnitude more efficient than the initially developed algorithms. A final, but significant, contribution is the identification of flaws in simple methods of generating random QCSP instances, and a means of generating instances which are not known to be flawed.     

Solving set-valued constraint satisfaction problems

In this paper, we consider the resolution of constraint satisfaction problems in the case where the variables of the problem are subsets of ${\mathbb{R}^{n}}$ . In order to use a constraint propagation approach, we introduce set intervals (named i-sets), which are sets of subsets of ${\mathbb{R}^{n}}$ with a lower bound and an upper bound with respect to the inclusion. Then, we propose basic operations for i-sets. This makes possible to build contractors that are then used by the propagation to solve problem involving sets as unknown variables. In order to illustrate the principle and the efficiency of the approach, a testcase is provided.     

A new approach to partial constraint satisfaction problems

A finite binary Constraint Satisfaction Problem (CSPs) is defined as consisting of a set of n problem variables, a domain of d potential values for each variable and a set of m binary constraints involving only two variables at a time. A solution to such a CSP is specified by assignment of a value to each variable that does not violate any of the constraints. The CSPs belong to the class of NP-Complete Problems. Backtracking and its variants have been generally used for solving CSPs. The class of Partial Constraint Satisfaction Problems (PCSPs) is a subclass of CSPs that are either too difficult to solve or are unsolvable. Near optimal solutions are always desired to these problems.

In this article, we have considered only finite binary CSPs or PCSPs and developed a method of time complexity O(n ² d ²) to obtain a near optimal solution for them. The performance of the method in terms of the average number of consistency checks and the average number of constraint violations is measured on various randomly generated binary CSPs and compared with the Branch and Bound (BB) method used to obtain the same solution. The BB method is a widely used optimization technique that may be viewed as a variation of backtracking. Thus, it was a natural choice in seeking an analog of backtracking to find optimal partial solutions for PCSPs. The proposed method moves much faster to the solution. The performance results indicate that in terms of the number of consistency checks, the proposed method has much less consistency checks than BB whereas in terms of average number of constraint violations both methods are same. An upper bound on the distance of the solution from the optimal solution is obtained analytically as ?n(n???2)(d???2)/(d???1)?.     

Relaxations of semiring constraint satisfaction problems

The Semiring Constraint Satisfaction Problem (SCSP) framework is a popular approach for the representation of partial constraint satisfaction problems. In this framework preferences can be associated with tuples of values of the variable domains. Bistarelli et al. [S. Bistarelli, U. Montanari, F. Rossi, Semiring-based constraint solving and optimization, Journal of the ACM 44 (2) (1997) 201-236] defines a maximal solution to a SCSP as the best set of solution tuples for the variables in the problem. Sometimes this maximal solution may not be good enough, and in this case we want to change the constraints so that we solve a problem that is slightly different from the original problem but has an acceptable solution. We propose a relaxation of a SCSP, and use a semiring to give a measure of the difference between the original SCSP and the relaxed SCSP. We introduce a relaxation scheme but do not address the computational aspects.     

Dynamic constraint satisfaction problems over models

In early phases of designing complex systems, models are not sufficiently detailed to serve as an input for automated synthesis tools. Instead, a design space is constituted by multiple models representing different valid design candidates. Design space exploration aims at searching through these candidates defined in the design space to find solutions that satisfy the structural and numeric design constraints and provide a balanced choice with respect to various quality metrics. Design space exploration in an model-driven engineering (MDE) context is frequently tackled as specific sort of constraint satisfaction problem (CSP). In CSP, declarative constraints capture restrictions over variables with finite domains where both the number of variables and their domains are required to be a priori finite. However, the existing formulation of constraint satisfaction problems can be too restrictive to capture design space exploration in many MDE applications with complex structural constraints expressed over the underlying models. In this paper, we interpret flexible and dynamic constraint satisfaction problems directly in the context of models. These extensions allow the relaxation of constraints during a solving process and address problems that are subject to change and require incremental re-evaluation. Furthermore, we present our prototype constraint solver for the domain of graph models built upon the Viatra2 model transformation framework and provide an evaluation of its performance with comparison to related tools.     

Backjump-based backtracking for constraint satisfaction problems

The performance of backtracking algorithms for solving finite-domain constraint satisfaction problems can be improved substantially by look-back and look-ahead methods. Look-back techniques extract information by analyzing failing search paths that are terminated by dead-ends. Look-ahead techniques use constraint propagation algorithms to avoid such dead-ends altogether. This paper describes a number of look-back variants including backjumping and constraint recording which recognize and avoid some unnecessary explorations of the search space. The last portion of the paper gives an overview of look-ahead methods such as forward checking and dynamic variable ordering, and discusses their combination with backjumping.     

Argumentation frameworks as constraint satisfaction problems

Argumentation is a promising approach for defeasible reasoning. It consists of justifying each plausible conclusion by arguments. Since the available information may be inconsistent, a conclusion and its negation may both be justified. The arguments are thus said to be conflicting. The main issue is how to evaluate the arguments. Several semantics were proposed for that purpose. The most important ones are: stable, preferred, complete, grounded and admissible. A semantics is a set of criteria that should be satisfied by a set of arguments, called extension, in order to be acceptable. Different decision problems related to these semantics were defined (like whether an argumentation framework has a stable extension). It was also shown that most of these problems are intractable. Consequently, developing algorithms for these problems is not trivial and thus the implementation of argumentation systems not obvious. Recently, some solutions to this problem were found. The idea is to use a reduction method where a given problem is translated in another one like SAT or ASP. This paper follows this line of research. It studies how to encode the problem of computing the extensions of an argumentation framework (under each of the previous semantics) as a constraint satisfaction problem (CSP). Such encoding is of great importance since it makes it possible to use the very efficient solvers (developed by the CSP community) for computing the extensions. Our encodings take advantage of existing reductions to SAT problems in the case of Dung’s abstract framework. Among the various ways of translating a SAT problem into a CSP one, we propose the most appropriate one in the argumentation context. We also provide encodings in case two other families of argumentation frameworks: the constrained version of Dung’s abstract framework and preference-based argumentation framework.     

STCSP — structured temporal constraint satisfaction problems

Temporal Constraint Satisfaction Problems (TCSP) is a well-known approach for representing and processing temporal knowledge. Important properties of the knowledge can be inferred by computing the minimal networks of TCSPs. Consistency and feasible values are immediately obtained; computing solutions can be assisted. Yet, in general, computing the minimal network of a disjunctive TCSP is intractable. The minimal network approach requires computation of the full network in order to answer a query. In this paper we characterize TCSPs for which subsets of the minimal network can be computed without having to compute the whole network. The partial computation is enabled by decomposition of the problem into a tree of sub-problems that share at most pairs of time points. Such decompositions are termed sim/2-tree decompositions. For TCSPs that have sim/2-tree decompositions, minimal constraints of input propositions can be computed by independent computations of the minimal networks of the sub-problems at most twice. It is also shown that the sim/2-tree characterization is a minimal set of conditions. The sim/2-tree decomposition extends former results about decomposition of a TCSP into bi-connected components. An algorithm for identifying a sim/2-tree decomposition of a TCSP is provided as well. Finally, the sim/2-tree decomposition is generalized in an inductive manner, which enables components of a decomposition to be further decomposed. For that purpose a model of Structured Temporal Constraint Satisfaction Problems (STCSP⁽ⁿ⁾, 0 ⩽ n), where STCSP⁽⁰⁾ is simply TCSP, STCSP⁽¹⁾ is a set of STCSP⁽⁰⁾s, and in general, STCSP⁽ⁿ⁾ for 1 ⩽ n is a set of STCSP^{(n − 1)}s, is introduced. This revised version was published online in June 2006 with corrections to the Cover Date.     

A knowledge representation for constraint satisfaction problems

We present a general representation for problems that can be reduced to constraint satisfaction problems (CSP) and a model for reasoning about their solution. The novel part of the model is a constraint-driven reasoner that manages a set of constraints specified in terms of arbitrarily complex Boolean expressions and represented in the form of a dependency network. This dependency network incorporates control information (derived from the syntax of the constraints) that is used for constraint propagation, contains dependency information that can be used for explanation and for dependency-directed backtracking, and is incremental in the sense that if the problem specification is modified, a new solution can be derived by modifying the existing solution. The constraint-driven reasoner is coupled to a problem solver which contains information about the problem variables and preference orderings     

Nogood-FC for solving partitionable constraint satisfaction problems

Many real problems can be naturally modelled as constraint satisfaction problems (CSPs). However, some of these problems are of a distributed nature, which requires problems of this kind to be modelled as distributed constraint satisfaction problems (DCSPs). In this work, we present a distributed model for solving CSPs. Our technique carries out a partition over the constraint network using a graph partitioning software; after partitioning, each sub-CSP is arranged into a DFS-tree CSP structure that is used as a hierarchy of communication by our distributed algorithm. We show that our distributed algorithm outperforms well-known centralized algorithms solving partitionable CSPs.     

Solving constraint satisfaction problems using hybrid evolutionarysearch

We combine the concept of evolutionary search with the systematic search concepts of arc revision and hill climbing to form a hybrid system that quickly finds solutions to static and dynamic constraint satisfaction problems (CSPs). Furthermore, we present the results of two experiments. In the first experiment, we show that our evolutionary hybrid outperforms a well-known hill climber, the iterative descent method (IDM), on a test suite of 750 randomly generated static CSPs. These results show the existence of a “mushy region” which contains a phase transition between CSPs that are based on constraint networks that have one or more solutions and those based on networks that have no solution. In the second experiment, we use a test suite of 250 additional randomly generated CSPs to compare two approaches for solving CSPs. In the first method, all the constraints of a CSP are known by the hybrid at run-time. We refer to this method as the static method for solving CSPs. In the second method, only half of the constraints of a CSPs are known at run-time. Each time that our hybrid system discovers a solution that satisfies all of the constraints of the current network, one additional constraint is added. This process of incrementally adding constraints is continued until all the constraints of a CSP are known by the algorithm or until the maximum number of individuals has been created. We refer to this second method as the dynamic method for solving CSPs. Our results show hybrid evolutionary search performs exceptionally well in the presence of dynamic (incremental) constraints, then also illuminate a potential hazard with solving dynamic CSPs