A strong local consistency for constraint satisfaction

Trends in Applied Intelligent Systems, 2010

Arc-consistency algorithms are widely used to prune the search space of Constraint Satisfaction Problems (CSPs). One of the most well-known arc-consistency algorithms for filtering CSPs is AC3. This algorithm repeatedly carries out revisions and requires support checks for identifying and deleting all unsupported values from the domains. Nevertheless, many revisions are ineffective, that is, they cannot delete any value and they require a lot of checks and are time-consuming. We present AC3-OP, an optimized and reformulated version of AC3 that reduces the number of constraint checks and prunes the same CSP search space with arithmetic constraints. In inequality constraints, AC3-OP, checks the binary constraints in both directions (full arc-consistency), but it only propagates new constraints in one direction. Thus, it avoids checking redundant constraints that do not filter any value of the variable's domain. The evaluation section shows the improvement of AC3-OP over AC3 in random instances.

2009

Dual Consistency (DC) was introduced by Lecoutre, Cardon and Vion in [10, 11]. DC is a novel way of handling Path Consistency (PC), with a simpler definition, and new efficient algorithms and approximations. Interestingly, the new definition may be extended to non-binary constraint networks (CNs). DC is thus a way to generalize PC to any CN, while keeping the initial non-binary constraints of the CN, and their associated propagators, untouched. DC can also be seen as a simple and efficient way to generate automatically implicit binary constraints. This article presents the implications of this generalization in terms of complexity. Preliminary experimental results shows the potential effectiveness of dynamic implicit constraints generation, as well as identifying its weaknesses. Prospective ideas of approximations, whose purpose is to handle these weaknesses in practice, are then proposed. Consistencies are properties of Constraint Networks (CN) that can be used to identify and prun...

Constraints, 2015

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 [23]. 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. * Some results included in this paper first appeared in . † This author has been funded by the EU project ICON (FP7-284715). 1 faster support search (e.g., and). Simple Tabular Reduction (STR) and its refinements maintain dynamically the support tables by removing invalid tuples from them during search. A recent approach holds information about removed values in the propagation queue and utilizes it to speed up support search . Given that GAC is a property defined on individual constraints, algorithms for GAC operate on one constraint at a time trying to filter infeasible values from the variables of the constraint. A different line of research has investigated stronger consistencies and algorithms to enforce them. Some of them are domain filtering, meaning that they only prune values from the domains of variables, e.g., see , whereas a few other ones are higher-order (or relation filtering), e.g., see , indicating that inconsistent tuples of values (nogoods of size 2 or more) can be identified. In contrast to GAC algorithms, the proposed algorithms to enforce these stronger consistencies are able to consider several constraints simultaneously. For example, pairwise consistency (PWC) is a relation filtering consistency that considers intersections between pairs of constraints. Recently there has been renewed interest for strong domain or relation filtering local consistencies as new ones have been proposed or/and efficient algorithms for existing ones have been devised . One of the most promising such consistencies is Max Restricted Pairwise Consistency (maxRPWC) , which is local consistency that is based PWC but can only make value deletions. In practice, strong consistencies are mainly applicable on constraints that are extensionally defined since intensionally defined constraints usually have specific semantics and are provided with efficient specialized filtering algorithms. However, a significant shortcoming of existing works on maxRPWC and other strong local consistencies is that the proposed algorithms for them are generic. That is, like earlier GAC algorithms such as GAC2001/3.1 [6], they are designed to operate on both intensional and extensional constraints, failing to recognize that strong consistencies are predominantly applicable on extensional constraints and should thus focus on such constraints. Despite the wealth of research on strong consistencies, they have not been widely adopted by CP solvers. State-of-the art solvers such as Gecode, Abscon, Choco, Minion, etc. predominantly apply GAC, and lesser forms of consistency such as bounds consistency, when propagating constraints. Regarding table constraints, CP solvers typically offer one or more of the above mentioned GAC methods for propagation. In this paper we propose filtering algorithms that achieve stronger consistency properties than GAC and have been specifically designed for table constraints. We contribute to both directions of domain and relation filtering methods by extending existing GAC algorithms for table constraints. The proposed methods are a step towards the efficient handling of intersecting table constraints and also provide specialization of strong local consistencies to extensional constraints that can be useful in practice. The first algorithm, called maxRPWC+, extends the GAC algorithm for table constraints of Lecoutre and Szymanek [23] and specializes the generic maxRWPC algorithm maxRPWC1 . The proposed domain filtering algorithm incorporates several techniques that help alleviate redundancies (i.e., redundant constraint checks and other operations on data structures) displayed by existing maxRPWC algorithms. We also describe a variant of maxRPWC+ which is more efficient when applied during search due to the lighter use of data structures.

2012

Local consistency properties and algorithms for enforcing them are central to the success of Constraint Processing. In this paper, we explore how to exploit the structure of the problem on the performance of the algorithm for enforcing consistency. We propose various strategies for managing the propagation queue of an algorithm for enforcing consistency, and empirically compare their effectiveness for solving CSPs with backtrack search and full lookahead. We focus our investigations on consistency algorithms that operate on the dual graph of a CSP and demonstrate the importance of exploiting a tree decomposition of the dual graph. Further, we note that exploiting structure is particularly striking on unsatisfiable instances. We conjecture that the approach for queue-management strategies benefits virtually all other propagation algorithms.

Constraints, 2011

Max Restricted Path Consistency (maxRPC) is a local consistency for binary constraints that enforces a higher order of consistency than arc consistency. Despite the strong pruning that can be achieved, maxRPC is rarely used because existing maxRPC algorithms suffer from overheads and redundancies as they can repeatedly perform many constraint checks without triggering any value deletions. In this paper we propose and evaluate techniques that can boost the performance of maxRPC algorithms by eliminating many of these overheads and redundancies. These include the combined use of two data structures to avoid many redundant constraint checks, and the exploitation of residues to quickly verify the existence of supports. Based on these, we propose a number of closely related maxRPC algorithms. The first one, maxRPC3, has optimal O(end 3 ) time complexity, displays good performance when used stand-alone, but is expensive to apply during search. The second one, maxRPC3 rm , has O(en 2 d 4 ) time complexity, but a restricted version with O(end 4 ) complexity can be very efficient when used during search. The other algorithms are simple modifications of maxRPC3 rm . All algorithms have O(ed) space complexity when used stand-alone. However, maxRPC3 has O(end) space This paper is an extended version of [1] that appeared in the proceedings of CP-2010.

Journal of the ACM, 1997

Constraint networks are a simple representation and reasoning framework with diverse applications. In this paper, we identify two new complementary properties on the restrictiveness of the constraints in a networkconstraint tightness and constraint looseness-and we show their usefulness for estimating the level of local consistency needed to ensure global consistency, and for estimating the level of local consistency present in a network. In particular, we present a sufficient condition, based on constraint tightness and the level of local consistency, that guarantees that a solution can be found in a backtrack-free manner. The condition can be useful in applications where a knowledge base will be queried over and over and the preprocessing costs can be amortized over many queries. We also present a sufficient condition for local consistency, based on constraint looseness, that is straightforward and inexpensive to determine. The condition can be used to estimate the level of local consistency of a network. This in turn can be used in deciding whether it would be useful to preprocess the network before a backtracking search, and in deciding which local consistency conditions, if any, still need to be enforced if we want to ensure that a solution can be found in a backtrack-free manner. Two definitions of local consistency are employed in characterizing the conditions: the traditional variable-based notion and a recently introduced definition of local consistency called relational consistency 1 .

2009

Abstract Powerful consistency techniques, such as AC* and FDAC*, have been developed for Weighted Constraint Satisfaction Problems (WCSPs) to reduce the space in solution search, but are restricted to only unary and binary constraints. On the other hand, van Hoeve et al. developed efficient graph-based algorithms for handling soft constraints as classical constraint optimization problems.

2017 IEEE 29th International Conference on Tools with Artificial Intelligence (ICTAI), 2017

Partial singleton (weak) path consistency, or partial ♦-consistency, for a qualitative constraint network, ensures that the process of instantiating any constraint of that network with any of its base relations b and enforcing partial (weak) path consistency, or partial ⋄-consistency, in the updated network, yields a partially ⋄-consistent subnetwork where the respective constraint is still defined by b. This local consistency is essential for helping to decide the satisfiability of challenging qualitative constraint networks and has been shown to play a crucial role in tackling more demanding problems associated with a given qualitative constraint network, such as the problem of minimal labeling. One of the main downsides to using partial ♦-consistency, is that it is computationally expensive to enforce in a given qualitative constraint network, as, despite being a local consistency in principle, it retains a global scope of the network at hand. In this paper, we propose a lazy alg...

1997

Filtering techniques are essential in order to ef ciently solve constraint satisfaction problems CSPs A blind search often leads to a com binatorial explosion the algorithm repeatedly nding the same local inconsistencies Main taining a local consistency can strongly reduce the search e ort especially on hard and large problems A good illustration are the good time performances on such problems of maintaining arc consistency during search compared to for ward checking which maintains a lower level of local consistency On the one hand arc consist ency consistency is the most used ltering technique because it cheaply removes some val ues that cannot belong to any solution On the other hand other k consistencies k have important space and time requirements because they can change the set of constraints They can only be used on very small CSPs Thus in this paper we study and compare the ltering techniques that are more pruningful than arc consistency while leaving unchanged the set of co...

2005

The GAC-Scheme has become a popular general purpose algorithm for solving n-ary constraints, although it may scan an exponential number of supporting tuples. In this paper, we develop a major improvement of this scheme. When searching for a support, our new algorithm is able to skip over a number of tuples exponential in the arity of the constraint by exploiting knowledge about the current domains of the variables. We demonstrate the effectiveness of the method for large table constraints.