Strong local consistency algorithms for table constraints

Table constraints are important in constraint programming as they are present in many real problems from areas such as configuration and databases. As a result, numerous specialized algorithms that achieve generalized arc consistency (GAC) on table constraints have been proposed. Since these algorithms achieve GAC, they operate on one constraint at a time. In this paper we propose new filtering algorithms for positive table constraints that achieve stronger local consistency properties than GAC by exploiting intersections between constraints. The first algorithm, called maxRPWC+, is a domain filtering algorithm that is based on the local consistency maxRPWC and extends the GAC algorithm of Lecoutre and Szymanek (2006). The second algorithm extends the state-of-the-art STR-based algorithms to stronger relation filtering consistencies, i.e., consistencies that can remove tuples from constraints’ relations. Experimental results from benchmark problems demonstrate that the proposed algorithms are quite competitive with standard GAC algorithms like STR2 in some classes of problems with intersecting table constraints, being orders of magnitude faster in some cases.     

The Complexity of Reasoning with Global Constraints

Constraint propagation is one of the techniques central to the success of constraint programming. To reduce search, fast algorithms associated with each constraint prune the domains of variables. With global (or non-binary) constraints, the cost of such propagation may be much greater than the quadratic cost for binary constraints. We therefore study the computational complexity of reasoning with global constraints. We first characterise a number of important questions related to constraint propagation. We show that such questions are intractable in general, and identify dependencies between the tractability and intractability of the different questions. We then demonstrate how the tools of computational complexity can be used in the design and analysis of specific global constraints. In particular, we illustrate how computational complexity can be used to determine when a lesser level of local consistency should be enforced, when constraints can be safely generalized, when decomposing constraints will reduce the amount of pruning, and when combining constraints is tractable.     

New filtering algorithms for combinations of among constraints

Several combinatorial problems, such as car sequencing and rostering, feature sequence constraints, restricting the number of occurrences of certain values in every subsequence of a given length. We present three new filtering algorithms for the sequence constraint, including the first that establishes domain consistency in polynomial time. The filtering algorithms have complementary strengths: One borrows ideas from dynamic programming; another reformulates it as a regular constraint; the last is customized. The last two algorithms establish domain consistency, and the customized one does so in polynomial time. We provide experimental results that demonstrate the practical usefulness of each. We also show that the customized algorithm applies naturally to a generalized version of the sequence constraint that allows subsequences of varied lengths. The significant computational advantage of using a single generalized sequence constraint over a semantically equivalent collection of among or sequence constraints is demonstrated empirically.     

Triangle-based consistencies for cost function networks

Cost Function Networks (aka Weighted CSP) allow to model a variety of problems, such as optimization of deterministic and stochastic graphical models including Markov random Fields and Bayesian Networks. Solving cost function networks is thus an important problem for deterministic and probabilistic reasoning. This paper focuses on local consistencies which define essential tools to simplify Cost Function Networks, and provide lower bounds on their optimal solution cost. To strengthen arc consistency bounds, we follow the idea of triangle-based domain consistencies for hard constraint networks (path inverse consistency, restricted or max-restricted path consistencies), describe their systematic extension to cost function networks, study their relative strengths, define enforcing algorithms, and experiment with them on a large set of benchmark problems. On some of these problems, our improved lower bounds seem necessary to solve them.     

CSPs with counters: a likelihood-based heuristic

Abstract. Counter constraints are a naturalrepresentation of constraints on the finite capacity of resources in resource-allocation type problems. They are a generic family of non-binary constraints that limit the number of variables that may be assigned particular values. Counter constraints can be represented by binary constraints, at a cost. We analyse the cost, show how a counter can be represented as a linear number of binary constraints, and demonstrate empirically that even with the optimal reduction,an explicit representation of counters is preferable to their representation as a set of binary constraints. For counter constraints, value ordering is essential. An heuristic for value ordering on constraint satisfaction problems (CSP), based on the estimated likelihoodof a solution, is presented. The proposed value ordering heuristic is useful for counter constraints, as well as for binary CSPs, where it can be used to approximate the number of solutions consistent with a particular value assignment to a variable. The proposed value ordering heuristic integrates counter constraints with binary constraint networks in a novel manner. Counter constraints are problematic for most heuristics, which are local in scope, yet we demonstrated empirically that the proposed value ordering heuristic is significantly superior to heuristics used in previous work.     

基于多核CPU的表约束并行传播模式研究

并行传播是并行约束程序领域中的一个研究方向,其研究内容是如何并行执行在约束上的过滤算法.根据维持表约束网络广义弧相容（generalized arc consistency,简称GAC）的串行传播模式,提出了维持表约束网络临时广义弧相容（temporary generalized arc consistency,简称TGAC）的并行传播模式,该模式基于多核CPU,由并行传播算法和并行过滤算法两部分组成;之后,给出了并行传播模式的可靠性证明,而且通过对并行传播模式的最坏时间复杂度分析,可以认为并行传播模式在平均过滤时间较长的实例上要快于串行传播模式;最终的实验结果也验证了上述结论,并行传播模式在多数实例集上取得了从1.4~3.4不等的加速比.     

一种基于时间戳的简单表缩减算法

表约束是一种外延的知识表示方法,每个约束在对应的变量集上列举出所有支持或禁止的元组.广义弧相容（generalized arc consistency,简称GAC）是求解约束满足问题应用最广泛的相容性.Simple Tabular Reduction（STR）是一类高效的维持GAC的算法.在回溯搜索中,STR动态地删除无效元组,降低了查找支持的开销,并拥有单位时间的回溯代价,在高元表约束上获得了广泛运用,并有大量基于STR的改进算法被提出,其中,元组集的压缩表示是目前研究较多的方法.同样基于动态维持元组集有效部分的思想,为STR提出一种检测并删除无效元组和为变量更新支持的算法,作用于原始表约束并拥有单位时间的回溯代价.实验结果表明,该算法在表约束上维持GAC的效率普遍高于现有的非基于压缩表示的STR算法,并且在一些实例上的效率高于最新的基于元组集压缩表示的STR算法.     

一种高效的FDE并行传播算法

约束规划(constraint programming, CP)是表示和求解组合问题的经典范式之一.扩展约束(extensional constraint)或称表约束(table constraint)是约束规划中最为常见的约束类型.绝大多数约束规划问题都可以用表约束表达.在问题求解时,相容性算法用于缩减搜索空间.目前,最为高效的表约束相容性算法是简单表约缩减(simple table reduction, STR)算法簇,如Compact-Table (CT)和STRbit算法.它们在搜索过程中维持广义弧相容(generalized arc consistency, GAC).此外,完全成对相容性(full pairwise consistency, fPWC)是一种比GAC剪枝能力更强的相容性.最为高效的维持fPWC算法是PW-CT算法.多年来,人们提出了多种表约束相容性算法来提高剪枝能力和执行效率.因子分解编码(factor-decomposition encoding, FDE)通过对平凡问题重新编码.它一定程度地扩大了问题模型,使在新的问题上维持相对较弱的GAC等价于在原问题...     

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].     

Extending GENET with lazy arc consistency

Many important applications, such as graph coloring, scheduling and production planning, can be solved by GENET, a local search method which is used to solve binary constraint satisfaction problems (CSPs). Where complete search methods are typically augmented with consistency methods to reduce the search, local search methods are not. We propose a consistency technique, lazy arc consistency, which is suitable for use within GENET. We show it can improve the efficiency of the GENET search on some instances of binary CSPs, and does not suffer the overhead of full arc consistency     

Tractable classes for CSPs of arbitrary arity: from theory to practice

The research of this thesis focuses on the analysis of polynomial classes and their practical exploitation for solving constraint satisfaction problems (CSPs) with finite domains. In particular, I worked on bridging the gap between theoretical works and practical results in constraint solvers. Specifically, the goal of this thesis is to find explanation for the effectiveness of solvers, and also to show that studied tractable classes are not artificial since several real-problems among the ones used in the CSP 2008 Competition belong to them.Our work is organized into three main parts. In the first part, we proposed several types of microstructures for CSPs of arbitrary arity which are based on some knwon binary encoding of non-binary CSPs like, dual encoding, hidden-variable transformation and mixed (or double) encoding. These theoretical tools are designed to facilitate the study of tractable classes, sets of CSP instances which can be solved in polytime, when the constraints are non-binary. After that, we propose a new tractable classes of CSPs whose the highlighting should allow on the one hand to explain the effectiveness of solvers of the state of the art namely FC, MAC, RFL and on the second hand to provide the opportunities for easy integration in these solvers. These would include the definition of new tractable classes without using of an ad hoc algorithms as in the traditional case. These new tractable classes are related to the number of maximal cliques in the microstructure of binary or non-binary CSP. In the last part, we focus on the presence of instances belonging to polynomial classes in classical benchmarks used by the CP community. We study in particular the Broken-Triangle Property (BTP) and its extension DBTP to CSP of arbitrary arity. Next, we prove that BTP can also be used to reduce the size of the search space by merging pairs of values on which no broken triangle exists. Finally, we introduce a formal framework, called transformation, and we develop the concept of hidden tractable class that we exploit from an experimental point of view.     

基于关联约束非二元弧一致性的约束满足问题求解

弧一致性算法在二元约束满足问题中取得了成功的应用,但并不能被有效泛化至预处理非二元约束满足问题(NCSP).本文提出了处理NCSP的关联约束非二元弧一致性算法.通过随机NCSP生成器产生问题实例,分别采用关联约束非二元弧一致性算法和非二元弧一致性算法进行预处理,并对预处理后的问题实例应用回溯算法进行求解.对比分析采用两种预处理算法和不采用预处理下回溯算法的求解性能,仿真实验结果表明关联约束非二元弧一致性算法可以有效地剔除冗余的约束元组和变量域值,使关联约束非二元弧一致性回溯算法具有更良好的鲁棒性.     

优化简单表缩减算法求解因子分解编码实例

表约束在约束程序（constraint programming,简称CP）中被广泛研究.目前,求解表约束问题效率最高的算法是CT （compact-table）和STRbit （simple tabular reduction bit）.它们在搜索过程中维持广义弧相容（generalized arc consistency,简称GAC）.完全成对相容（full pairwise consistency,简称fPWC）是一种强于GAC的相容性关系,目前,实现fPWC效率最高的算法是PW-CT,但是它无法直接在通用的求解器上实现.因子分解编码（factor-decomposition encoding,简称FDE）是实现fPWC的一种编码方式,通常和简单表缩减（STR）算法一起来使用.当前效率最高的STR算法使用了bitset的数据结构,用这些算法来求解FDE实例可能会造成内存溢出.提出了STRFDE算法——一种使用bitset结构来求解FDE实例的方法.它结合了CT和STRbit的优势,在保证求解效率的同时,使占用的内存尽可能小.实验结果表明,在许多存在非平凡相交的实例上,该算法是有竞争力的.     

Theoretical analysis of singleton arc consistency and its extensions

Singleton arc consistency (SAC) is a consistency property that is simple to specify and is stronger than arc consistency. Algorithms have already been proposed to enforce SAC, but they have a high time complexity. In this paper, we give a lower bound to the worst-case time complexity of enforcing SAC on binary constraints. We discuss two interesting features of SAC. The first feature leads us to propose an algorithm for SAC that has optimal time complexity when restricted to binary constraints. The second feature leads us to extend SAC to a stronger level of local consistency that we call Bidirectional SAC (BiSAC). We also show the relationship between SAC and the propagation of disjunctive constraints.     

A unified framework for consistency check

Constraint Satisfaction Problem (CSP) involves finding values for variables to satisfy a set of constraints. Consistency check is the key technique in solving this class of problems. Past research has developed many algorithms for such a purpose, e.g., node consistency, are consistency, generalized node and arc consistency, specific methods for checking specific constraints, etc. In this article, an attempt is made to unify these algorithms into a common framework. This framework consists of two parts. the first part is a generic consistency check algorithm, which allows and encourages each individual constraint to be checked by its specific consistency methods. Such an approach provides a direct way of practical implementation of the CSP model for real problem-solving. the second part is a general schema for describing the handling of each type of constraint. the schema characterizes various issues of constraint handling in constraint satisfaction, and provides a common language for expressing, discussing, and exchanging constraint handling techniques. © 1995 John Wiley & Sons, Inc.     

Relational consistency algorithms and their application in finding subgraph and graph isomorphisms

The determination of subgraph and graph isomorphisms is an important application for the algebraic manipulation of networks of binary constraints. Simplified and streamlined arc consistency and tree search algorithms are introduced, and experimental results show substantial reduction in timings compared with previous algorithms for determining isomorphisms. Several path consistency algorithms, including a new one, have been timed experimentally on isomorphism problems, and found not to be cost effective despite their theoretical appeal. The importance of this result is enhanced by the absence of previously published experimentation with path consistency. A theoretical study of the new path consistency algorithm provides insight into the experimental results.     

SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems

This paper introduces a surrogate model based algorithm for computationally expensive mixed-integer black-box global optimization problems with both binary and non-binary integer variables that may have computationally expensive constraints. The goal is to find accurate solutions with relatively few function evaluations. A radial basis function surrogate model (response surface) is used to select candidates for integer and continuous decision variable points at which the computationally expensive objective and constraint functions are to be evaluated. In every iteration multiple new points are selected based on different methods, and the function evaluations are done in parallel. The algorithm converges to the global optimum almost surely. The performance of this new algorithm, SO-MI, is compared to a branch and bound algorithm for nonlinear problems, a genetic algorithm, and the NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) algorithm for mixed-integer problems on 16 test problems from the literature (constrained, unconstrained, unimodal and multimodal problems), as well as on two application problems arising from structural optimization, and three application problems from optimal reliability design. The numerical experiments show that SO-MI reaches significantly better results than the other algorithms when the number of function evaluations is very restricted (200–300 evaluations).     

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.     

Approximated consistency for the automatic recording constraint

We introduce the automatic recording constraint (ARC) that can be used to model and solve scheduling problems where tasks may not overlap in time and the tasks linearly exhaust some resource. Since achieving generalized arc-consistency for the ARC is NP-hard, we develop a filtering algorithm that achieves approximated consistency only. Numerical results show the benefits of the new constraint on three out of four different types of benchmark sets for the automatic recording problem. On these instances, run-times can be achieved that are orders of magnitude better than those of the best previous constraint programming approach.     

Redundant modeling in permutation weighted constraint satisfaction problems

In classical constraint satisfaction, redundant modeling has been shown effective in increasing constraint propagation and reducing search space for many problem instances. In this paper, we investigate, for the first time, how to benefit the same from redundant modeling in weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and over-constrained problems. Our work focuses on a popular and special class of problems, namely, permutation problems. First, we show how to automatically generate a redundant permutation WCSP model from an existing permutation WCSP using generalized model induction. We then uncover why naively combining mutually redundant permutation WCSPs by posting channeling constraints as hard constraints and relying on the standard node consistency (NC*) and arc consistency (AC*) algorithms would miss pruning opportunities, which are available even in a single model. Based on these observations, we suggest two approaches to handle the combined WCSP models. In our first approach, we propose m\text -NC_{\text c}^*m\text {-NC}_{\text c}^* and m\text -AC_{\text c}^*m\text {-AC}_{\text c}^* and their associated algorithms for effectively enforcing node and arc consistencies in a combined model with m sub-models. The two notions are strictly stronger than NC* and AC* respectively. While the first approach specifically refines NC* and AC* so as to apply to combined models, in our second approach, we propose a parameterized local consistency LB(m,Φ). The consistency can be instantiated with any local consistency Φ for single models and applied to a combined model with m sub-models. We also provide a simple algorithm to enforce LB(m,Φ). With the two suggested approaches, we demonstrate their applicabilities on several permutation problems in the experiments. Prototype implementations of our proposed algorithms confirm that applying 2\text -NC_{\text c}^*,  2\text -AC_{\text c}^*2\text {-NC}_{\text c}^*,\;2\text {-AC}_{\text c}^*, and LB(2,Φ) on combined models allow far more constraint propagation than applying the state-of-the-art AC*, FDAC*, and EDAC* algorithms on single models of hard benchmark problems.