How to Determine the Expressive Power of Constraints

Some constraint languages are more powerful than others because they allow us to express a larger collection of problems. In this paper, we give a precise meaning to this concept of expressive power for constraints over finite sets of values. The central result of the paper is that the expressive power of a given set of constraint types is determined by certain algebraic properties of the underlying relations. These algebraic properties can be calculated by solving a particular constraint satisfaction problem, which we call an 'indicator problem'. We discuss the connection between expressive power and computational complexity, and show that indicator problems provide a simple method to test for tractability.     

General Properties and Termination Conditions for Soft Constraint Propagation

Soft constraints based on semirings are a generalization of classical constraints, where tuples of variables' values in each soft constraint are associated to elements from an algebraic structure called semiring. This framework is able to express, for example, fuzzy, classical, weighted, valued and over-constrained constraint problems.Classical constraint propagation has been extended and adapted to soft constraints by defining a schema for soft constraint propagation [8]. On the other hand, in [1–3] it has been proven that most of the well known constraint propagation algorithms for classical constraints can be cast within a single schema.In this paper we combine these two schemas and we provide a more general framework where the schema of [3] can be used for soft constraints. In doing so, we generalize the concept of soft constraint propagation, and we provide new sufficient and independent conditions for its termination.     

Constraints and universal algebra

In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two relational structures. We give a number of examples to illustrate how this framework can be used to express a wide variety of combinatorial problems, many of which are not generally considered as constraint satisfaction problems. We also show that certain key aspects of the mathematical structure of constraint satisfaction problems can be precisely described in terms of the notion of a Galois connection, which is a standard notion of universal algebra. Using this result, we obtain an algebraic characterisation of the property of minimality in a constraint satisfaction problem. We also obtain a similar algebraic criterion for determining whether or not a given set of solutions can be expressed by a constraint satisfaction problem with a given structure, or a given set of allowed constraint types. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Implementing a Test for Tractability

The question of determining which sets of constraints give rise to NP-complete problems, and which give rise to tractable problems, is an important open problem in the theory of constraint satisfaction. It has been shown in previous papers that certain sufficient conditions for tractability and NP-completeness can be identified using algebraic properties of relations, and that these conditions can be tested by solving a particular form of constraint satisfaction problem (the so-called indicator problem).This paper describes a program which can solve the relevant indicator problems for arbitrary sets of constraints over small domains, and for some sets of constraints over larger domains. The main innovation in the program is its ability to deal with the many symmetries present in the problem; it also has the ability to preserve symmetries in cases where this speeds up the solution.Using this program, we have systematically investigated the complexity of all individual binary relations over a domain of size four or less, and of all individual ternary relations over a domain of size three or less. This automated analysis includes the derivation of more than 450 000 new NP-completeness results, and precisely identifies the small set of individual relations which cannot be classified as either tractable or NP-complete using the algebraic conditions presented in previous papers.     

Constraint Retraction in CLP(FD): Formal Framework and Performance Results

Constraint retraction can be described, in general, as the possibility of deleting a previously stated piece of information. This is obviously very convenient in many programming frameworks, especially in those that involve some level of interaction between the user and the system, or also in those concerning rescheduling or replanning. Nevertheless, constraint retraction is usually not provided in current constraint programming environments. This is mainly due to its high complexity and also to its non-monotonic nature, which would make most of such systems much more complex to reason with. In this paper we avoid these problems by considering a specific constraint programming framework, called clpFD, that is, constraint logic programming (CLP) over finite domain (FD) constraints. We propose an algorithm which deletes a constraint from a set of FD constraints, while maintaining partial arc-consistency, which is usual in this programming framework. What is crucial is that the retraction operation we propose is incremental, in that it follows the chain of dependencies among variables which are set by the nature of the FD constraints, and by doing so it updates only the part of the constraint set which is affected by the deletion. We also detail how constraint retraction can be incorporated in the FD constraint solver and we evaluate its behavior within the clpFD system. Experimental results on usual benchmarks, on classes of problems of increasing connectivity, and also on a real-life problem show that in almost all cases the use of our retraction algorithm provides great speed-up with respect to standard methods while not slowing down the clpFD system when no retraction is performed. This provides the system with an efficient way of retracting constraints while not changing its performance when the user does not want to use this new feature.     

MiniBrass: Soft constraints for MiniZinc

Over-constrained problems are ubiquitous in real-world decision and optimization problems. Plenty of modeling formalisms for various problem domains involving soft constraints have been proposed, such as weighted, fuzzy, or probabilistic constraints. All of them were shown to be instances of algebraic structures. In terms of modeling languages, however, the field of soft constraints lags behind the state of the art in classical constraint optimization. We introduce MiniBrass, a versatile soft constraint modeling language building on the unifying algebraic framework of partially ordered valuation structures (PVS) that is implemented as an extension of MiniZinc and MiniSearch. We first demonstrate the adequacy of PVS to naturally augment partial orders with a combination operation as used in soft constraints. Moreover, we provide the most general construction of a c-semiring from an arbitrary PVS. Both arguments draw upon elements from category theory. MiniBrass turns these theoretical considerations into practice: It offers a generic extensible PVS type system, reusable implementations of specific soft constraint formalisms as PVS types, operators for complex PVS products, and morphisms to transform PVS. MiniBrass models are compiled into MiniZinc to benefit from the wide range of solvers supporting FlatZinc. We evaluated MiniBrass on 28 “softened” MiniZinc benchmark problems with six different solvers. The results demonstrate the feasibility of our approach.     

Bases for Boolean co-clones

The complexity of various problems in connection with Boolean constraints, like, for example, quantified Boolean constraint satisfaction, have been studied recently. Depending on what types of constraints may be used, the complexity of such problems varies. A very interesting observation of the recent past has been that the thus derived classification of constraints can be explained with the help of universal algebra. More precisely, the difficulty of such a constraint problem often depends on the co-clone the constraints are from. A co-clone is a set of Boolean relations that is closed under very natural closure operations. Nearly all these co-clones can be generated by said operators out of a finite set of relations, a so-called base. Knowing a, preferably simple, base for each co-clone can therefore be of great value when studying the complexity of Boolean constraint problems, since this knowledge reduces the infinitely many cases of equivalent problems to a single one—the constraint satisfaction problem for this base. In this paper we give a finite and simple base for every Boolean co-clone, where this is possible. We give evidence that the presented bases are as easy as possible.     

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.