Solving finite domain constraint hierarchies by local consistency and tree search

We provide a reformulation of the constraint hierarchies (CHs) framework based on the notion of error indicators. Adapting the generalised view of local consistency in semiring-based constraint satisfaction problems, we define constraint hierarchy k-consistency (CH-k-C) and give a CH-2-C enforcement algorithm. We demonstrate how the CH-2-C algorithm can be seamlessly integrated into the ordinary branch-and-bound algorithm to make it a finite domain (FD) CH solver. Experimentation confirms the efficiency and robustness of our proposed solver prototype. Unlike other FD CH solvers, our proposed method works for both local and global comparators. In addition, our solver can support arbitrary error functions.     

多值传播的相容性技术

相容性技术是求解约束满足问题的重要手段. 本文针对目前已有相容性算法的单值传播特点, 提出多值传播理论, 证明出k次单值传播与一次多值传播的等价性, 在此基础上, 给出多值传播的弧相容定理. 将该定理与目前流行的Singleton弧相容技术结合, 得到多值传播算法SAC-MP, 并证明其完备性和正确性. 通过对随机问题、N皇后、鸽巢问题及基准用例的测试表明, 算法SAC-MP的执行效率是已有算法SAC-SDS和SAC-3的2～3倍.     

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.     

Explaining circuit propagation

The circuit constraint is used to constrain a graph represented by a successor for each node, such that the resulting edges form a circuit. Circuit and its variants are important for various kinds of tour-finding, path-finding and graph problems. In this paper we examine how to integrate the circuit constraint, and its variants, into a lazy clause generation solver. To do so we must extend the constraint to explain its propagation. We consider various propagation algorithms for circuit and examine how best to explain each of them. We compare the effectiveness of each propagation algorithm once we use explanation, since adding explanation changes the trade-off between propagation complexity and power. Simpler propagators, although less powerful, may produce more reusable explanations. Even though the most powerful propagator considered for circuit and variants creates huge explanations, we find that explanation is highly advantageous for solving problems involving this kind of constraint.     

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...     

View-based propagation of decomposable constraints

Constraints that may be obtained by composition from simpler constraints are present, in some way or another, in almost every constraint program. The decomposition of such constraints is a standard technique for obtaining an adequate propagation algorithm from a combination of propagators designed for simpler constraints. The decomposition approach is appealing in several ways. Firstly because creating a specific propagator for every constraint is clearly infeasible since the number of constraints is infinite. Secondly, because designing a propagation algorithm for complex constraints can be very challenging. Finally, reusing existing propagators allows to reduce the size of code to be developed and maintained. Traditionally, constraint solvers automatically decompose constraints into simpler ones using additional auxiliary variables and propagators, or expect the users to perform such decomposition themselves, eventually leading to the same propagation model. In this paper we explore views, an alternative way to create efficient propagators for such constraints in a modular, simple and correct way, which avoids the introduction of auxiliary variables and propagators.     

最大受限路径相容约束传播算法的研究进展

约束传播技术对于约束满足问题的求解性能至关重要。约束传播技术在一个预处理过程中能彻底地移除一些局部不相容值,或者在搜索期间高效地剪枝搜索树。最大受限路径相容算法(max Restricted Path Consistency,maxRPC)是最近提出的一种强相容性约束传播算法,它能够删除更多不相容值,在解决复杂问题中取得了很好的效果。文中对弧相容算法AC和最大受限路径相容算法maxRPC的相关算法AC3,AC3rm,maxRPC1,maxRPC2,maxRPCrm,maxRPC3等及其相关变体分别进行介绍和比较。在Mistral求解器上的实验测试结果验证了各种算法的性能。     

The dilemma between arc and bounds consistency

Consistency enforcement is used to prune the search tree of a constraint satisfaction problem (CSP). Arc consistency (AC) is a well-studied consistency level, with many implementations. Bounds consistency (BC), a looser consistency level, is known to have equal time complexity to AC. To solve a CSP, we have to implement an algorithm of our own or employ an existing solver. In any case, at some point, we have to decide between enforcing either AC or BC. As the choice between AC or BC is more or less predefined and currently made without considering the individualities of each CSP, this study attempts to make this decision deterministic and efficient, without the need of trial and error. We find that BC fits better while solving a CSP with its maximum domains' size being greater than its constrained variables number. We study the behavior of maintaining arc or bounds consistency during search, and we show how the overall search methods complexity is affected by the employed consistency level.     

High-Order Consistency in Valued Constraint Satisfaction

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.     

View-based propagator derivation

When implementing a propagator for a constraint, one must decide about variants: When implementing min, should one also implement max? Should one implement linear constraints both with unit and non-unit coefficients? Constraint variants are ubiquitous: implementing them requires considerable (if not prohibitive) effort and decreases maintainability, but will deliver better performance than resorting to constraint decomposition. This paper shows how to use views to derive propagator variants, combining the efficiency of dedicated propagator implementations with the simplicity and effortlessness of decomposition. A model for views and derived propagators is introduced. Derived propagators are proved to be perfect in that they inherit essential properties such as correctness and domain and bounds consistency. Techniques for systematically deriving propagators such as transformation, generalization, specialization, and type conversion are developed. The paper introduces an implementation architecture for views that is independent of the underlying constraint programming system. A detailed evaluation of views implemented in Gecode shows that derived propagators are efficient and that views often incur no overhead. Views have proven essential for implementing Gecode, substantially reducing the amount of code that needs to be written and maintained.     

一种新的基于完全独立相容性的预处理技术

研究了求解约束满足问题(Constraint satisfaction problem, CSP)中的预处理技术. 首先提出了子论域上的完全独立相容性(Entirety singleton consistency, ESC)概念和相应算法, 分析并证明了算法的复杂性和正确性, 而后对其两条重要性质进行了详细证明. 基于此概念和性质, 提出了一种基于完全独立相容性的预处理算法: SAC-ESC算法, 并给出了正确性证明. 最后, 本文采用分治思想, 根据不同问题的论域自适应地合理划分其子论域. 实验结果表明, 对于随机问题、鸽巢问题、N皇后问题和基准用例, 算法SAC-ESC的执行效率大约是目前流行算法SAC-SDS和SAC-3的3~20倍.     

Parallel consistent labeling algorithms

Mackworth and Freuder have analyzed the time complexity of several constraint satisfaction algorithms.⁽¹⁾ Mohr and Henderson have given new algorithms, AC-4 and PC-3, for arc and path consistency, respectively, and have shown that the arc consistency algorithm is optimal in time complexity and of the same order space complexity as the earlier algorithms.⁽²⁾ In this paper, we give parallel algorithms for solving node and arc consistency. We show that any parallel algorithm for enforcing are consistency in the worst case must have O(na) sequential steps, wheren is number of nodes, anda is the number of labels per node. We give several parallel algorithms to do arc consistency. It is also shown that they all have optimal time complexity. The results of running the parallel algorithms on a BBN Butterfly multiprocessor are also presented.This work was partially supported by NSF Grants MCS-8221750, DCR-8506393, and DMC-8502115.     

The power of propagation: when GAC is enough

Considerable effort in constraint programming has focused on the development of efficient propagators for individual constraints. In this paper, we consider the combined power of such propagators when applied to collections of more than one constraint. In particular we identify classes of constraint problems where such propagators can decide the existence of a solution on their own, without the need for any additional search. Sporadic examples of such classes have previously been identified, including classes based on restricting the structure of the problem, restricting the constraint types, and some hybrid examples. However, there has previously been no unifying approach which characterises all of these classes: structural, language-based and hybrid. In this paper we develop such a unifying approach and embed all the known classes into a common framework. We then use this framework to identify a further class of problems that can be solved by propagation alone.     

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.     

From soft constraints to bipolar preferences: modelling framework and solving issues

Real-life problems present several kinds of preferences. We focus on problems with both positive and negative preferences, which we call bipolar preference problems. Although seemingly specular notions, these two kinds of preferences should be dealt with differently to obtain the desired natural behaviour. We technically address this by generalising the soft constraint formalism, which is able to model problems with one kind of preference. We show that soft constraints model only negative preferences, and we add to them a new mathematical structure which allows to handle positive preferences as well. We also address the issue of the compensation between positive and negative preferences, studying the properties of this operation. Finally, we extend the notion of arc consistency to bipolar problems, and we show how branch and bound (with or without constraint propagation) can be easily adapted to solve such problems.     

最先失败原则的约束传播算法

约束满足问题是人工智能研究领域的重要问题.而弧相容算法是求解约束满足问题的重要工具.在弧相容算法中应用启发式规则已经证明是一种很有效的方式.本文提出一个基于最先失败原则的约束传播算法,该算法在搜索过程中更早地发现含有空域的变量并提前进行回溯,从而提高问题求解效率.同时,在"明月1.0"架构下实现了该算法,实验结果表明使用最先失败原则的弧相容算法要比原来的算法效率上提高了约40%.     

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.     

Experimental Analysis of Numeric and Symbolic Constraint Satisfaction Techniques for Temporal Reasoning

Many temporal applications like planning and scheduling can be viewed as special cases of the numeric and symbolic temporal constraint satisfaction problem. Thus we have developed a temporal model, TemPro, based on the interval Algebra, to express such applications in term of qualitative and quantitative temporal constraints. TemPro extends the interval algebra relations of Allen to handle numeric information. To solve a constraint satisfaction problem, different approaches have been developed. These approaches generally use constraint propagation to simplify the original problem and backtracking to directly search for possible solutions. The constraint propagation can also be used during the backtracking to improve the performance of the search. The objective of this paper is to assess different policies for finding if a TemPro network is consistent. The main question we want to answer here is how much constraint propagation is useful for finding a single solution for a TemPro constraint graph. For this purpose, we have experimented by randomly generating large consistent networks for which either arc and/or path consistency algorithms (AC-3, AC-7 and PC-2) were applied. The main result of this study is an optimal policy combining these algorithms either at the symbolic (Allen relation propagation) or at the numerical level.     

An optimal coarse-grained arc consistency algorithm

The use of constraint propagation is the main feature of any constraint solver. It is thus of prime importance to manage the propagation in an efficient and effective fashion. There are two classes of propagation algorithms for general constraints: fine-grained algorithms where the removal of a value for a variable will be propagated to the corresponding values for other variables, and coarse-grained algorithms where the removal of a value will be propagated to the related variables. One big advantage of coarse-grained algorithms, like AC-3, over fine-grained algorithms, like AC-4, is the ease of integration when implementing an algorithm in a constraint solver. However, fine-grained algorithms usually have optimal worst case time complexity while coarse-grained algorithms do not. For example, AC-3 is an algorithm with non-optimal worst case complexity although it is simple, efficient in practice, and widely used. In this paper we propose a coarse-grained algorithm, AC2001/3.1, that is worst case optimal and preserves as much as possible the ease of its integration into a solver (no heavy data structure to be maintained during search). Experimental results show that AC2001/3.1 is competitive with the best fine-grained algorithms such as AC-6. The idea behind the new algorithm can immediately be applied to obtain a path consistency algorithm that has the best-known time and space complexity. The same idea is then extended to non-binary constraints.     

Probabilistic Reasoning Under Coherence in System P

In this paper we apply a probabilistic reasoning under coherence to System P. We consider a notion of strict probabilistic consistency, we show its equivalence to Adams' probabilistic consistency, and we give a necessary and sufficient condition for probabilistic entailment. We consider the inference rules of System P in the framework of coherent imprecise probabilistic assessments. Exploiting our coherence-based approach, we propagate the lower and upper probability bounds associated with the conditional assertions of a given knowledge base, obtaining the precise probability bounds for the derived conclusions of the inference rules. This allows a more flexible and realistic use of System P in default reasoning and provides an exact illustration of the degradation of the inference rules when interpreted in probabilistic terms. We also examine the disjunctive Weak Rational Monotony rule of System P⁺ proposed by Adams in his extended probabilistic logic. Finally, we examine the propagation of lower bounds with real -values and, to illustrate our probabilistic reasoning, we consider an example.