STR2: optimized simple tabular reduction for table constraints

Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints and/or to compress their representation. In this paper, we describe an optimization of simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamically maintain the tables of supports when generalized arc consistency (GAC) is enforced/maintained. STR2, the new refined GAC algorithm we propose, allows us to limit the number of operations related to validity checking and search of supports. Interestingly enough, this optimization makes simple tabular reduction potentially r times faster where r is the arity of the constraint(s). The results of an extensive experimentation that we have conducted with respect to random and structured instances indicate that STR2 is usually around twice as fast as the original STR, two or three times faster than the approach based on the hidden variable encoding, and can be up to one order of magnitude faster than previously state-of-the-art (generic) GAC algorithms on some series of instances. When comparing STR2 with the more recently developed algorithm based on multi-valued decision diagrams (MDDs), we show that both approaches are rather complementary.     

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.     

Constraint hierarchies

Constraints allow programmers and users to state declaratively a relation that should be maintained, rather than requiring them to write procedures to maintain the relation themselves. They are thus useful in such applications as programming languages, user interface toolkits, and simulation packages. In many situations, it is desirable to be able to state bothrequired andpreferential constraints. The required constraints must hold. Since the other constraints are merely preferences, the system should try to satisfy them if possible, but no error condition arises if it cannot. Aconstraint hierarchy consists of a set of constraints, each labeled as either required or preferred at some strength. An arbitrary number of different strengths is allowed. In the discussion of a theory of constraint hierarchies, we present alternate ways of selecting among competing possible solutions, and prove a number of propositions about the relations among these alternatives. We then outline algorithms for satisfying constraint hierarchies, and ways in which we have used constraint hierarchies in a number of programming languages and systems.     

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.     

Dynamic bottleneck optimization for k-edge and 2-vertex connectivity

We consider the problem of updating efficiently the minimum value b over a weighted graph, so that edges with a cost less than b induce a spanning subgraph satisfying a k-edge or 2-vertex connectivity constraint, when the cost of an edge of the graph is updated. Our results include update algorithms of almost linear time (up to poly-logarithmic factors) in the number of vertices for all considered connectivity constraints (for fixed k), and a worst case construction showing that these algorithms are almost optimal in their class.     

Constraint hierarchies

Constraints allow programmers and users to state declaratively a relation that should be maintained, rather than requiring them to write procedures to maintain the relation themselves. They are thus useful in such applications as programming languages, user interface toolkits, and simulation packages. In many situations, it is desirable to be able to state bothrequired andpreferential constraints. The required constraints must hold. Since the other constraints are merely preferences, the system should try to satisfy them if possible, but no error condition arises if it cannot. Aconstraint hierarchy consists of a set of constraints, each labeled as either required or preferred at some strength. An arbitrary number of different strengths is allowed. In the discussion of a theory of constraint hierarchies, we present alternate ways of selecting among competing possible solutions, and prove a number of propositions about the relations among these alternatives. We then outline algorithms for satisfying constraint hierarchies, and ways in which we have used constraint hierarchies in a number of programming languages and systems.     

Ordering and Path Constraints over Semistructured Data

Constraints are a valuable tool for managing information. Feature constraints have been used for describing records in constraint programming (Aït-Kaci and Podelski, 1993; Smolka and Treinen, 1994) and record like structures in computational linguistics (Kaplan and Bresnan, 1982; Shieber, 1986). In this paper, we consider how constraint-based technology can be used to query and reason about semistructured data. The constraint system FT (Müller et al., 1997) provides information ordering constraints interpreted over feature trees. Here, we show how a generalization of FT combined with path constraints can be used to formally represent, state constraints, and reason about semistructured data. The constraint languages we propose provide possibilities to straightforwardly capture, for example, what it means for a tree to be a subtree or subsumed by another, or what it means for two paths to be divergent. We establish a logical semantics for our constraints thanks to axiom schemes presenting our first-order theory constraint system. We propose using the constraint systems for querying semistructured data.     

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.     

An optimal arc consistency algorithm for a particular case of sequence constraint

The AtMostSeqCard constraint is the conjunction of a cardinality constraint on a sequence of n variables and of n???q?+?1 constraints AtMost u on each subsequence of size q. This constraint is useful in car-sequencing and crew-rostering problems. In van Hoeve et al. (Constraints 14(2):273–292, 2009), two algorithms designed for the AmongSeq constraint were adapted to this constraint with an O(2 ^q n) and O(n ³) worst case time complexity, respectively. In Maher et al. (2008), another algorithm similarly adaptable to filter the AtMostSeqCard constraint with a time complexity of O(n ²) was proposed. In this paper, we introduce an algorithm for achieving arc consistency on the AtMostSeqCard constraint with an O(n) (hence optimal) worst case time complexity. Next, we show that this algorithm can be easily modified to achieve arc consistency on some extensions of this constraint. In particular, the conjunction of a set of m AtMostSeqCard constraints sharing the same scope can be filtered in O(nm). We then empirically study the efficiency of our propagator on instances of the car-sequencing and crew-rostering problems.     

A characterization of convex problems in decentralized Control/sup */

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

A characterization of convex problems in decentralized control

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

Automatic image segmentation using constraint learning and propagation

In this paper, we propose automatic image segmentation using constraint learning and propagation. Recently, kernel learning is receiving much attention because a learned kernel can fit the given data better than a predefined kernel. To effectively learn the constraints generated by initial seeds for image segmentation, we employ kernel propagation (KP) based on kernel learning. The key idea of KP is first to learn a small-sized seed-kernel matrix and then propagate it into a large-sized full-kernel matrix. By applying KP to automatic image segmentation, we design a novel segmentation method to achieve high performance. First, we generate pairwise constraints, i.e., must-link and cannot-link, from initially selected seeds to make the seed-kernel matrix. To select the optimal initial seeds, we utilize global k-means clustering (GKM) and self-tuning spectral clustering (SSC). Next, we propagate the seed-kernel matrix into the full-kernel matrix of the entire image, and thus image segmentation results are obtained. We test our method on the Berkeley segmentation database, and the experimental results demonstrate that the proposed method is very effective in automatic image segmentation.     

Itemset mining: A constraint programming perspective

The field of data mining has become accustomed to specifying constraints on patterns of interest. A large number of systems and techniques has been developed for solving such constraint-based mining problems, especially for mining itemsets. The approach taken in the field of data mining contrasts with the constraint programming principles developed within the artificial intelligence community. While most data mining research focuses on algorithmic issues and aims at developing highly optimized and scalable implementations that are tailored towards specific tasks, constraint programming employs a more declarative approach. The emphasis lies on developing high-level modeling languages and general solvers that specify what the problem is, rather than outlining how a solution should be computed, yet are powerful enough to be used across a wide variety of applications and application domains.This paper contributes a declarative constraint programming approach to data mining. More specifically, we show that it is possible to employ off-the-shelf constraint programming techniques for modeling and solving a wide variety of constraint-based itemset mining tasks, such as frequent, closed, discriminative, and cost-based itemset mining. In particular, we develop a basic constraint programming model for specifying frequent itemsets and show that this model can easily be extended to realize the other settings. This contrasts with typical procedural data mining systems where the underlying procedures need to be modified in order to accommodate new types of constraint, or novel combinations thereof. Even though the performance of state-of-the-art data mining systems outperforms that of the constraint programming approach on some standard tasks, we also show that there exist problems where the constraint programming approach leads to significant performance improvements over state-of-the-art methods in data mining and as well as to new insights into the underlying data mining problems. Many such insights can be obtained by relating the underlying search algorithms of data mining and constraint programming systems to one another. We discuss a number of interesting new research questions and challenges raised by the declarative constraint programming approach to data mining.     

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.     

Managing dynamic CSPs with preferences

We present a new framework, managing Constraint Satisfaction Problems (CSPs) with preferences in a dynamic environment. Unlike the existing CSP models managing one form of preferences, ours supports four types, namely: unary and binary constraint preferences, composite preferences and conditional preferences. This offers more expressive power in representing a wide variety of dynamic constraint applications under preferences and where the possible changes are known and available a priori. Conditional preferences allow some preference functions to be added dynamically to the problem, during the resolution process, if a given condition on some variables is true. A composite preference is a higher level of preference among the choices of a composite variable. Composite variables are variables whose possible values are CSP variables. In other words, this allows us to represent disjunctive CSP variables. The preferences are viewed as a set of soft constraints using the fuzzy CSP framework. Solving constraint problems with preferences consists in finding a solution satisfying all the constraints while optimizing the global preference value. This is handled by four variants of the branch and bound algorithm, we propose in this paper, and where constraint propagation is used to improve the time efficiency in practice. In order to evaluate and compare the performance of these four strategies, we conducted an experimental study on randomly generated dynamic CSPs with quantitative preferences. The results are reported and discussed in the paper.     

Constraint-based timetabling a case study

In this article we concentrate on a typical scheduling problem: the computation of a timetable for a German college. Like many other scheduling problems, this problem contains a variety of complex constraints and necessitates special-purpose search strategies. Techniques from operations research and traditional constraint logic programming are not able to express these constraints and search strategies on a sufficiently high level of abstraction. We show that the higher order concurrent constraint language Oz provides this high-level expressivity, and can serve as a useful programming tool for college time-tabling.     

Topology optimization with displacement constraints: a bilevel programming approach

We consider the minimum-compliance formulation of the truss topology problem with additional linear constraints on the displacements: the so-called displacement constraints. We propose a new bilevel programming approach to this problem. Our primal goal (upper-level) is to satisfy the displacement constraint as well as possible — we minimize the gap between the actual and prescribed displacement. Our second goal (lower level) is to minimize the compliance — we still want to find the stiffest structure satisfying the displacement constraints. On the lower level we solve a standard truss topology problem and hence we can solve it in the formulation suitable for the fast interior point alogrithms. The overall bilevel problem is solved by means of the so-called implicit programming approach. This approach leads to a nonsmooth optimization problem which is finally solved by a nonsmooth solver.     

Local and global symmetry breaking in itemset mining

The concept of symmetry has been extensively studied in the field of constraint programming and in the propositional satisfiability. Several methods for detection and removal of these symmetries have been developed, and their use in known solvers of these domains improved dramatically their effectiveness on a big variety of problems considered difficult to solve. The concept of symmetry may be exported to other areas where some structures can be exploited effectively. Particularly, in the area of data mining where some tasks can be expressed as constraints or logical formulas. We are interested here, by the detection and elimination of local and global symmetries in the item-set mining problem. Recent works have provided effective encodings as Boolean constraints for these data mining tasks and some idea on symmetry elimination in this area begin to appear, but still few and the techniques presented are often on global symmetry that is detected and eliminated statically in a preprocessing phase. In this work we study the notion of local symmetry and compare it to global symmetry for the itemset mining problem. We show how local symmetries of the boolean encoding can be detected dynamically and give some properties that allow to eliminate theses symmetries in SAT-based itemset mining solvers in order to enhance their efficiency.     

Other things besides number: Abstraction,constraint propagation,and string variable types

In constraint programming (CP), a combinatorial problem is modeled declaratively as a conjunction of constraints, each of which captures some of the combinatorial substructure of the problem. Constraints are more than a modeling convenience: every constraint is partially implemented by an inference algorithm, called a propagator, that rules out some but not necessarily all infeasible candidate values of one or more unknowns in the scope of the constraint. Interleaving propagation with systematic search leads to a powerful and complete solution method, combining a high degree of re-usability with natural, high-level modeling.A propagator can be characterized as a sound approximation of a constraint on an abstraction of sets of candidate values; propagators that share an abstraction are similar in the strength of the inference they perform when identifying infeasible candidate values. In this thesis, we consider abstractions of sets of candidate values that may be described by an elegant mathematical formalism, the Galois connection. We develop a theoretical framework from the correspondence between Galois connections and propagators, unifying two disparate views of the abstraction-propagation connection, namely the oft-overlooked distinction between representational and computational over-approximations. Our framework yields compact definitions of propagator strength, even in complicated cases (i.e., involving several types, or unknowns with internal structure); it also yields a method for the principled derivation of propagators from constraint definitions.We apply this framework to the extension of an existing CP solver to constraints over strings, that is, words of finite length. We define, via a Galois connection, an over-approximation for bounded-length strings, and demonstrate two different methods for implementing this over-approximation in a CP solver. First we use the Galois connection to derive a bounded-length string representation as an aggregation of existing scalar types; propagators for this representation are obtained by manual derivation, or automated synthesis, or a combination. Then we implement a string variable type, motivating design choices with knowledge gained from the construction of the over-approximation. The resulting CP solver extension not only substantially eases modeling for combinatorial string problems, but also leads to substantial efficiency improvements over prior CP methods.