Structural decompositions for problems with global constraints

A wide range of problems can be modelled as constraint satisfaction problems (CSPs), that is, a set of constraints that must be satisfied simultaneously. Constraints can either be represented extensionally, by explicitly listing allowed combinations of values, or implicitly, by special-purpose algorithms provided by a solver. Such implicitly represented constraints, known as global constraints, are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. In recent years, a variety of restrictions on the structure of CSP instances have been shown to yield tractable classes of CSPs. However, most such restrictions fail to guarantee tractability for CSPs with global constraints. We therefore study the applicability of structural restrictions to instances with such constraints. We show that when the number of solutions to a CSP instance is bounded in key parts of the problem, structural restrictions can be used to derive new tractable classes. Furthermore, we show that this result extends to combinations of instances drawn from known tractable classes, as well as to CSP instances where constraints assign costs to satisfying assignments.     

A note on some collapse results of valued constraints

Valued constraint satisfaction problem (VCSP) is an optimisation framework originally coming from Artificial Intelligence and generalising the classical constraint satisfaction problem (CSP). The VCSP is powerful enough to describe many important classes of problems. In order to investigate the complexity and expressive power of valued constraints, a number of algebraic tools have been developed in the literature. In this note we present alternative proofs of some known results without using the algebraic approach, but by representing valued constraints explicitly by combinations of other valued constraints.     

Constant Unary Constraints and Symmetric Real-Weighted Counting Constraint Satisfaction Problems

A unary constraint (on the Boolean domain) is a function from {0,1} to the set of real numbers. A free use of auxiliary unary constraints given besides input instances has proven to be useful in establishing a complete classification of the computational complexity of approximately solving weighted counting Boolean constraint satisfaction problems (or #CSPs). In particular, two special constant unary constraints are a key to an arity reduction of arbitrary constraints, sufficient for the desired classification. In an exact counting model, both constant unary constraints are always assumed to be available since they can be eliminated efficiently using an arbitrary nonempty set of constraints. In contrast, we demonstrate, in an approximate counting model, that at least one of them is efficiently approximated and thus eliminated approximately by a nonempty constraint set. This fact directly leads to an efficient construction of polynomial-time randomized approximation-preserving Turing reductions (or AP-reductions) from #CSPs with designated constraints to any given #CSPs composed of symmetric real-valued constraints of arbitrary arities even in the presence of arbitrary extra unary constraints.     

On the power of structural decompositions of graph-based representations of constraint problems

The Constraint Satisfaction Problem (CSP) is a central issue of research in Artificial Intelligence. Due to its intractability, many efforts have been made in order to identify tractable classes of CSP instances, and in fact deep and useful results have already been achieved. In particular, this paper focuses on structural decomposition methods, which are essentially meant to look for near-acyclicity properties of the graphs or hypergraphs that encode the structure of the constraints interactions. In general, constraint scopes comprise an arbitrary number of variables, and thus this structure may be naturally encoded via hypergraphs. However, in many practical applications, decomposition methods are applied over suitable graph representations of the (possibly non-binary) CSP instances at hand. Despite the great interest in such binary approaches, a formal analysis of their power, in terms of their ability of identifying islands of tractability, was missing in the literature.The aim of this paper is precisely to fill this gap, by studying the relationships among binary structural methods, and by providing a clear picture of the tractable fragments of CSP that can be specified with respect to each of these decomposition approaches, when they are applied to binary representations of non-binary CSP instances. In particular, various long-standing questions about primal, dual and incidence graph encodings are answered. The picture is then completed by comparing methods on binary encodings with methods specifically conceived for non-binary constraints.     

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.     

Symmetry Definitions for Constraint Satisfaction Problems

We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the k-ary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.     

Structural tractability of enumerating CSP solutions

The problem of deciding whether CSP instances admit solutions has been deeply studied in the literature, and several structural tractability results have been derived so far. However, constraint satisfaction comes in practice as a computation problem where the focus is either on finding one solution, or on enumerating all solutions, possibly projected to some given set of output variables. The paper investigates the structural tractability of the problem of enumerating (possibly projected) solutions, where tractability means here computable with polynomial delay (WPD), since in general exponentially many solutions may be computed. A framework based on the notion of tree projection of hypergraphs is considered, which generalizes all structural decomposition methods that are based on decomposing a given instance into suitable tree-like groups of polynomial-time computable subproblems. Tractability results have been obtained both for classes of structures where output variables are part of their specification, and for classes of structures where computability WPD must be ensured for any possible set of output variables. By exhibiting dichotomies, these results are shown to be tight for classes of structures having bounded arity.     

Tractable Decision for a Constraint Language Implies Tractable Search

A constraint satisfaction problem (CSP) instance has a set of variables, each of which can take values in some domain. It also has a set of constraints, each of which restricts the variables in its scope to take values limited by its constraint relation.The language of a constraint satisfaction problem instance is the set of different constraint relations used in its specification. For a given set of relations over some domain we define the problem CSP () to the set of CSP instances whose language is contained in .The decision problem for a set of CSP instances is, given an instance in the class, to decide whether a solution exists. The search problem is to find such a solution. Here we address the connection between the tractability of the decision and search problems. We prove that given a constraint language over a finite domain for which the decision problem for CSP () is tractable, the search problem is always tractable.We define a surjective language over a finite domain in a standard way. We also show that we can determine in polynomial time whether an instance over a surjective language with a tractable decision problem has fewer than k solutions, and that we can generate all of its solutions with polynomial delay.     

Maximum H-colourable subdigraphs and constraint optimization with arbitrary weights

In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of positive-weight constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so that the total weight of satisfied constraints is maximized. We consider this problem and its variant Max AW CSP where the weights are allowed to be both positive and negative, and study how the complexity of the problems depends on the allowed constraint types. We prove that Max AW CSP over an arbitrary finite domain exhibits a dichotomy: it is either polynomial-time solvable or NP-hard. Our proof builds on two results that may be of independent interest: one is that the problem of finding a maximum H-colourable subdigraph in a given digraph is either NP-hard or trivial depending on H, and the other a dichotomy result for Max CSP with a single allowed constraint type.     

Logic-based solution methods for optimal control of hybrid systems

Combinatorial optimization over continuous and integer variables is a useful tool for solving complex optimal control problems of hybrid dynamical systems formulated in discrete-time. Current approaches are based on mixed-integer linear (or quadratic) programming (MIP), which provides the solution after solving a sequence of relaxed linear (or quadratic) programs. MIP formulations require the translation of the discrete/logic part of the hybrid problem into mixed-integer inequalities. Although this operation can be done automatically, most of the original symbolic structure of the problem (e.g., transition functions of finite state machines, logic constraints, symbolic variables, etc.) is lost during the conversion, with a consequent loss of computational performance. In this paper, we attempt to overcome such a difficulty by combining numerical techniques for solving convex programming problems with symbolic techniques for solving constraint satisfaction problems (CSP). The resulting "hybrid" solver proposed here takes advantage of CSP solvers for dealing with satisfiability of logic constraints very efficiently. We propose a suitable model of the hybrid dynamics and a class of optimal control problems that embrace both symbolic and continuous variables/functions, and that are tailored to the use of the new hybrid solver. The superiority in terms of computational performance with respect to commercial MIP solvers is shown on a centralized supply chain management problem with uncertain forecast demand.     

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.     

Minimization of Locally Defined Submodular Functions by Optimal Soft Arc Consistency

Submodular function minimization is a polynomially solvable combinatorial problem. Unfortunately the best known general-purpose algorithms have high-order polynomial time complexity. In many applications the objective function is locally defined in that it is the sum of cost functions (also known as soft or valued constraints) whose arities are bounded by a constant. We prove that every valued constraint satisfaction problem with submodular cost functions has an equivalent instance on the same constraint scopes in which the actual minimum value of the objective function is rendered explicit. Such an equivalent instance is the result of establishing optimal soft arc consistency and can hence be found by solving a linear program. From a practical point of view, this provides us with an alternative algorithm for minimizing locally defined submodular functions. From a theoretical point of view, this brings to light a previously unknown connection between submodularity and soft arc consistency.     

Markov constraints: steerable generation of Markov sequences

Markov chains are a well known tool to model temporal properties of many phenomena, from text structure to fluctuations in economics. Because they are easy to generate, Markovian sequences, i.e. temporal sequences having the Markov property, are also used for content generation applications such as text or music generation that imitate a given style. However, Markov sequences are traditionally generated using greedy, left-to-right algorithms. While this approach is computationally cheap, it is fundamentally unsuited for interactive control. This paper addresses the issue of generating steerable Markovian sequences. We target interactive applications such as games, in which users want to control, through simple input devices, the way the system generates a Markovian sequence, such as a text, a musical sequence or a drawing. To this aim, we propose to revisit Markov sequence generation as a branch and bound constraint satisfaction problem (CSP). We propose a CSP formulation of the basic Markovian hypothesis as elementary Markov Constraints (EMC). We propose algorithms that achieve domain-consistency for the propagators of EMCs, in an event-based implementation of CSP. We show how EMCs can be combined to estimate the global Markovian probability of a whole sequence, and accommodate for different species of Markov generation such as fixed order, variable-order, or smoothing. Such a formulation, although more costly than traditional greedy generation algorithms, yields the immense advantage of being naturally steerable, since control specifications can be represented by arbitrary additional constraints, without any modification of the generation algorithm. We illustrate our approach on simple yet combinatorial chord sequence and melody generation problems and give some performance results.     

Solving constraint satisfaction problems using hybrid evolutionarysearch

We combine the concept of evolutionary search with the systematic search concepts of arc revision and hill climbing to form a hybrid system that quickly finds solutions to static and dynamic constraint satisfaction problems (CSPs). Furthermore, we present the results of two experiments. In the first experiment, we show that our evolutionary hybrid outperforms a well-known hill climber, the iterative descent method (IDM), on a test suite of 750 randomly generated static CSPs. These results show the existence of a “mushy region” which contains a phase transition between CSPs that are based on constraint networks that have one or more solutions and those based on networks that have no solution. In the second experiment, we use a test suite of 250 additional randomly generated CSPs to compare two approaches for solving CSPs. In the first method, all the constraints of a CSP are known by the hybrid at run-time. We refer to this method as the static method for solving CSPs. In the second method, only half of the constraints of a CSPs are known at run-time. Each time that our hybrid system discovers a solution that satisfies all of the constraints of the current network, one additional constraint is added. This process of incrementally adding constraints is continued until all the constraints of a CSP are known by the algorithm or until the maximum number of individuals has been created. We refer to this second method as the dynamic method for solving CSPs. Our results show hybrid evolutionary search performs exceptionally well in the presence of dynamic (incremental) constraints, then also illuminate a potential hazard with solving dynamic CSPs     

Relatively quantified constraint satisfaction

The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more general framework of quantified constraint satisfaction, in which variables can be quantified both universally and existentially. We study the relatively quantified constraint satisfaction problem (RQCSP), in which the values for each individual variable can be arbitrarily restricted. We give a complete complexity classification of the cases of the RQCSP where the types of constraints that may appear are specified by a constraint language.     

Comparing evolutionary algorithms on binary constraint satisfaction problems

Constraint handling is not straightforward in evolutionary algorithms (EAs) since the usual search operators, mutation and recombination, are 'blind' to constraints. Nevertheless, the issue is highly relevant, for many challenging problems involve constraints. Over the last decade, numerous EAs for solving constraint satisfaction problems (CSP) have been introduced and studied on various problems. The diversity of approaches and the variety of problems used to study the resulting algorithms prevents a fair and accurate comparison of these algorithms. This paper aligns related work by presenting a concise overview and an extensive performance comparison of all these EAs on a systematically generated test suite of random binary CSPs. The random problem instance generator is based on a theoretical model that fixes deficiencies of models and respective generators that have been formerly used in the evolutionary computing field.     

A hybrid algorithm for finding minimal unsatisfiable subsets in over‐constrained CSPs

Minimal Unsatisfiable Subsets (MUSes) are the subsets of constraints of an overconstrained constraint satisfaction problem (CSP) that cannot be satisfied simultaneously and therefore are responsible for the conflict in the CSP. In this paper, we present a hybrid algorithm for finding MUSes in overconstrained CSPs. The hybrid algorithm combines the direct and the indirect approaches to finding MUSes in overconstrained CSPs. Experimentation with random CSPs reveals that the hybrid approach is not only quite efficient but when operating under a time bound it finds a more representative set of MUSes. © 2011 Wiley Periodicals, Inc.     

Constraint satisfaction with bounded treewidth revisited

We consider the constraint satisfaction problem (CSP) parameterized by the treewidth of primal, dual, and incidence graphs, combined with several other basic parameters such as domain size and arity. We determine all combinations of the considered parameters that admit fixed-parameter tractability.     

Periodic Constraint Satisfaction Problems: Tractable Subclasses

We study a generalization of the constraint satisfaction problem (CSP), the periodic constraint satisfaction problem. An input instance of the periodic CSP is a finite set of generating constraints over a structured variable set that implicitly specifies a larger, possibly infinite set of constraints; the problem is to decide whether or not the larger set of constraints has a satisfying assignment. This model is natural for studying constraint networks consisting of constraints obeying a high degree of regularity or symmetry. Our main contribution is the identification of two broad polynomial-time tractable subclasses of the periodic CSP.