Solving quantified constraint satisfaction problems with value selection rules

Solving a quantified constraint satisfaction problem (QCSP) is usually a hard task due to its computational complexity. Exact algorithms play an important role in solving this problem, among which backtrack algorithms are effective. In a backtrack algorithm, an important step is assigning a variable by a chosen value when exploiting a branch, and thus a good value selection rule may speed up greatly. In this paper, we propose two value selection rules for existentially and universally quantified variables, respectively, to avoid unnecessary searching. The rule for universally quantified variables is prior to trying failure values in previous branches, and the rule for existentially quantified variables selects the promising values first. Two rules are integrated into the state-of-the-art QCSP solver, i.e., QCSPSolve, which is an exact solver based on backtracking. We perform a number of experiments to evaluate improvements brought by our rules. From computational results, we can conclude that the new value selection rules speed up the solver by 5 times on average and 30 times at most. We also show both rules perform well particularly on instances with existentially and universally quantified variables occurring alternatively.     

Existentially restricted quantified constraint satisfaction

The quantified constraint satisfaction problem (QCSP) is a framework for modelling PSPACE computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP.     

Relatively quantified constraint satisfaction

The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more general framework of quantified constraint satisfaction, in which variables can be quantified both universally and existentially. We study the relatively quantified constraint satisfaction problem (RQCSP), in which the values for each individual variable can be arbitrarily restricted. We give a complete complexity classification of the cases of the RQCSP where the types of constraints that may appear are specified by a constraint language.     

一种量词约束满足问题的混合易解子类

量词约束满足问题是人工智能和自动推理领域的一个重要问题.寻找多项式时间易解子类,是研究此类问题计算复杂性的关键.通过分析二元量词约束满足问题中的约束关系特征,以及量词前缀中的全称量词排列的顺序,提出了针对全称量词变量子结构的易解性质的分析方法.通过该方法,扩展了已知的基于Broken-Triangle Property的多项式时间易解子类,提出了一个更一般化的量词约束满足问题的混合易解子类.讨论了易解子类在问题结构分析中的一个应用,即通过易解子类确定量词约束满足问题的隐蔽变量集合,并通过实验分析不同易解子类所确定的集合大小.实验改造了基于回溯算法的求解器,在回溯过程中加入了易解子类的识别算法,并采用随机约束满足问题的生成模型作为测试基准.通过对比实验,验证了提出的多项式时间易解子类可以识别出更小的隐蔽变量集合,因此,新提出的易解子类在确定隐蔽变量集合方面更具优势.最后阐述了其他已有的混合易解子类也可以通过类似方法进行扩展,从而得到更多的一般化的理论结果.     

A Rigorous Global Filtering Algorithm for Quadratic Constraints*

This article introduces a new filtering algorithm for handling systems of quadratic equations and inequations. Such constraints are widely used to model distance relations in numerous application areas ranging from robotics to chemistry. Classical filtering algorithms are based upon local consistencies and thus, are often unable to achieve a significant pruning of the domains of the variables occurring in quadratic constraint systems. The drawback of these approaches comes from the fact that the constraints are handled independently. We introduce here a global filtering algorithm that works on a tight linear relaxation of the quadratic constraints. The Simplex algorithm is then used to narrow the domains. Since most implementations of the Simplex work with floating point numbers and thus, are unsafe, we provide a procedure to generate safe linearizations. We also exploit a procedure provided by Neumaier and Shcherbina to get a safe objective value when calling the Simplex algorithm. With these two procedures, we prevent the Simplex algorithm from removing any solution while filtering linear constraint systems. Experimental results on classical benchmarks show that this new algorithm yields a much more effective pruning of the domains than local consistency filtering algorithms.*This article is an extended version of [23].     

Complexity of Clausal Constraints Over Chains

We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x≥d and x≤d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NP-complete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature.     

On a decision procedure for quantified linear programs

Quantified linear programming is the problem of checking whether a polyhedron specified by a linear system of inequalities is non-empty, with respect to a specified quantifier string. Quantified linear programming subsumes traditional linear programming, since in traditional linear programming, all the program variables are existentially quantified (implicitly), whereas, in quantified linear programming, a program variable may be existentially quantified or universally quantified over a continuous range. In this paper, the term linear programming is used to describe the problem of checking whether a system of linear inequalities has a feasible solution. On account of the alternation of quantifiers in the specification of a quantified linear program (QLP), this problem is non-trivial. QLPs represent a class of declarative constraint logic programs (CLPs) that are extremely rich in their expressive power. The complexity of quantified linear programming for arbitrary constraint matrices is unknown. In this paper, we show that polynomial time decision procedures exist for the case in which the constraint matrix satisfies certain structural properties. We also provide a taxonomy of quantified linear programs, based on the structure of the quantifier string and discuss the computational complexities of the constituent classes. This research has been supported in part by the Air Force Office of Scientific Research under contract FA9550-06-1-0050.     

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.     

Non-binary quantified CSP: algorithms and modelling

The Quantified Constraint Satisfaction Problem (QCSP) extends classical CSP in a way which allows reasoning about uncertainty. In this paper I present novel algorithms for solving QCSP. Firstly I present algorithms to perform constraint propagation on reified disjunction constraints of any length. The algorithms make full use of quantifier information to provide a high level of consistency. Secondly I present a scheme to enforce the non-binary pure value rule. This rule is capable of pruning universal variables. Following this, two problems are modelled in non-binary QCSP: the game of Connect 4, and a variant of job-shop scheduling with uncertainty, in the form of machine faults. The job shop scheduling example incorporates probability bounding of scenarios (such that only fault scenarios above a probability threshold are considered) and optimization of the schedule makespan. These contribute to the art of modelling in QCSP, and are a proof of concept for applying QCSP methods to complex, realistic problems. Both models make use of the reified disjunction constraint, and the non-binary pure value rule. The example problems are used to evaluate the QCSP algorithms presented in this paper, identifying strengths and weaknesses, and to compare them to other QCSP approaches.     

Out of order quantifier elimination for Standard Quantified Linear Programs

In this paper, we present an out of order quantifier elimination algorithm for a class of Quantified Linear Programs (QLPs) called Standard Quantified Linear Programs (SQLPs). QLPs in general and SQLPs in particular are extremely useful constraint logic programming structures that find wide applicability in the modeling of real-time schedulability specifications; see Subramani [Subramani, K., 2005a. A comprehensive framework for specifying clairvoyance, constraints and periodicity in real-time scheduling. The Computer Journal 48 (3), 259–272]. Consequently any algorithmic advance in their solution has a strong practical impact. Prior to this work, the only known approaches to the solution of QLPs involved sequential variable elimination; see Subramani [Subramani, K., 2003b. An analysis of quantified linear programs. In: Calude, C.S. et al. (Eds.), Proceedings of the 4th International Conference on Discrete Mathematics and Theoretical Computer Science. DMTCS. In: Lecture Notes in Computer Science, vol. 2731. Springer-Verlag, pp. 265–277]. In the sequential approach, the innermost quantified variable is eliminated first, followed by the variable which then becomes the innermost quantified variable and so on, until we are left with a single variable from which the satisfiability of the original formula is easily deduced. This approach is applicable in both discrete and continuous domains; however, it is to be noted that the logic demanding the sequential approach requires that the variables are discrete-valued. To the best of our knowledge, the necessity for sequential elimination over continuous-valued variables has not been investigated in the literature. The techniques used in the development of our elimination algorithm may find applications in domains such as classical logic and finite model theory. The final aspect of our research concerns the structure-preserving nature of the algorithm that we introduce here; in general, it is not known whether discrete domains admit such elimination procedures.     

相接行凸约束网络的快速识别算法

约束网络为计算机科学中的许多问题提供了一种有效的表示方法.一般而言,约束满足问题是NP完全的.然而,许多实际问题通常对约束的结构或形式施加了特殊的限制,从而能够高效地加以解决.迄今,为了识别易处理约束类,人们对特殊的约束或约束网络方面进行了许多研究.相接行凸(connected row-convex,简称CRC)约束网络是Deville等人提出的一类易处理问题.为了给该类问题寻求有效的快速识别算法,在CRC约束网络相关工作基础上,提出了CRC约束矩阵的标准型.在分析CRC约束矩阵的标准型性质的基础上,利用行凸(row-convex,简称RC)约束网络的判定,结合PQ树(由P节点和Q节点构成的树)的性质和矩阵的索引表示法,给出了CRC约束网络的快速识别算法.该算法的时间复杂度为O(n³d²),其中,n为约束网络涉及的变量数,d为各变量的定义域中最大定义域的大小.该时间复杂度达到该类问题的最佳时间复杂度,从而为实际的CRC约束满足问题的求解提供了可行的方法.     

A tabu search heuristic for the quay crane scheduling problem

This paper proposes a tabu search heuristic for the Quay Crane Scheduling Problem (QCSP), the problem of scheduling a fixed number of quay cranes in order to load and unload containers into and from a ship. The optimality criterion considered is the minimum completion time. Precedence and non-simultaneity constraints between tasks are taken into account. The former originate from the different kind of operations that each crane has to perform; the latter are needed in order to avoid interferences between the cranes. The QCSP is decomposed into a routing problem and a scheduling problem. The routing problem is solved by a tabu search heuristic, while a local search technique is used to generate the solution of the scheduling problem. This is done by minimizing the longest path length in a disjunctive graph. The effectiveness of our algorithm is assessed by comparing it to a branch-and-cut algorithm and to a Greedy Randomized Adaptive Search Procedure (GRASP).     

Low-level dichotomy for quantified constraint satisfaction problems

Building on a result of Larose and Tesson for constraint satisfaction problems (CSPs), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP(B), where B is a finite structure that is a core. Specifically, such problems are either in ALogtime or are L-hard. This involves demonstrating that if CSP(B) is first-order expressible, and B is a core, then QCSP(B) is in ALogtime.We show that the class of B such that CSP(B) is first-order expressible (indeed trivial) is a microcosm for all QCSPs. Specifically, for any B there exists a C — generally not a core — such that CSP(C) is trivial, yet QCSP(B) and QCSP(C) are equivalent under logspace reductions.     

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.     

Optimizing an equilibrium supply chain network design problem by an improved hybrid biogeography based optimization algorithm

This paper presents a new equilibrium optimization method for supply chain network design (SCND) problem under uncertainty, where the uncertain transportation costs and customer demands are characterized by both probability and possibility distributions. We introduce cost risk level constraint and joint service level constraint in the proposed optimization model. When the random parameters follow normal distributions, we reduce the risk level constraint and the joint service level constraint into their equivalent credibility constraints. Furthermore, we employ a sequence of discrete possibility distributions to approximate continuous possibility distributions. To enhance solution efficiency, we introduce the dominance set and efficient valid inequalities into deterministic mixed-integer programming (MIP) model, and preprocess the valid inequalities to obtain a simplified nonlinear programming model. After that, a hybrid biogeography based optimization (BBO) algorithm incorporating new solution presentation and local search operations is designed to solve the simplified optimization model. Finally, we conduct some numerical experiments via an application example to demonstrate the effectiveness of the designed hybrid BBO.     

Integrating Standard Dependency Schemes in QCSP Solvers

Quantified constraint satisfaction problems (QCSPs) are an extension to constraint satisfaction problems (CSPs) with both universal quantifiers and existential quantifiers.In this paper we apply variab...     

Parametric Sequence Alignment with Constraints

Approximate matching techniques based on string alignment are important tools for investigating similarities between strings, such as those representing DNA and protein sequences. We propose a constraint based approach for parametric sequence alignment which allows for more general string alignment queries where the alignment cost can itself be parameterized as a query with some initial constraints. Thus, the costs need not be fixed in a parametric alignment query unlike the case in normal alignment. The basic dynamic programming string edit distance algorithm is generalized to a naive algorithm which uses inequalities to represent the alignment score. The naive algorithm is rather costly and the remainder of the paper develops an improvement which prunes alternatives where it can and approximates the alternatives otherwise. This reduces the number of inequalities significantly and strengthens the constraint representation with equalities. We present some preliminary results using parametric alignment on some general alignment queries.     

An MDD-based generalized arc consistency algorithm for positive and negative table constraints and some global constraints

A table constraint is explicitly represented as its set of solutions or non-solutions. This ad hoc (or extensional) representation may require space exponential to the arity of the constraint, making enforcing GAC expensive. In this paper, we address the space and time inefficiencies simultaneously by presenting the mddc constraint. mddc is a global constraint that represents its (non-)solutions with a multi-valued decision diagram (MDD). The MDD-based representation has the advantage that it can be exponentially smaller than a table. The associated GAC algorithm (called mddc) has time complexity linear to the size of the MDD, and achieves full incrementality in constant time. In addition, we show how to convert a positive or negative table constraint into an mddc constraint in time linear to the size of the table. Our experiments on structured problems, car sequencing and still-life, show that mddc is also a fast GAC algorithm for some global constraints such as sequence and regular. We also show that mddc is faster than the state-of-the-art generic GAC algorithms in Gent et al. (2007), Lecoutre and Szymanek (2006), Lhomme and Régin (2005) for table constraint.     

Propagation algorithms for lexicographic ordering constraints

Finite-domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finite-domain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pattern is the use of matrices of decision variables. The rows and/or columns of these matrices are often symmetric, leading to redundancy in a systematic search for solutions. An effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically. This paper develops an incremental propagation algorithm, GACLexLeq, that establishes generalised arc consistency on this constraint in O(n) operations, where n is the length of the vectors. Furthermore, this paper shows that decomposing GACLexLeq into primitive constraints available in current finite-domain constraint toolkits reduces the strength or increases the cost of constraint propagation. Also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering, detection of entailment, and vectors of unequal length. Experimental results on a number of domains demonstrate the value of GACLexLeq.     

Hybrid search for minimal perturbation in Dynamic CSPs

It is often the case that after a scheduling problem has been solved some small changes occur that make the solution of the original problem not valid. Solving the new problem from scratch can result in a schedule that is very different from the original schedule. In applications such as a university course timetable or flight scheduling, one would be interested in a solution that requires minimal changes for the users. The present paper considers the minimal perturbation problem. It is motivated by scenarios in which a Constraint Satisfaction Problem (CSP) is subject to changes. In particular, the case where some of the constraints are changed after a solution was obtained. The goal is to find a solution to the changed problem that is as similar as possible (e.g. includes minimal perturbations) to the previous solution. Previous studies proposed a formal model for this problem (Barták et al. 2004), a best first search algorithm (Ross et al. 2000), complexity bounds (Hebrard et al. 2005), and branch and bound based algorithms (Barták et al. 2004; Hebrard et al. 2005). The present paper proposes a new approach for solving the minimal perturbation problem. The proposed method interleaves constraint optimization and constraint satisfaction techniques. Our experimental results demonstrate the advantage of the proposed algorithm over former algorithms. Experiments were performed both on random CSPs and on random instances of the Meeting Scheduling Problem.