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.     

A branch and bound algorithm for numerical Max-CSP

The Constraint Satisfaction Problem (CSP) framework allows users to define problems in a declarative way, quite independently from the solving process. However, when the problem is over-constrained, the answer “no solution” is generally unsatisfactory. A Max-CSP \(\mathcal{P}_m = \langle V, \textbf{D}, C \rangle\) is a triple defining a constraint problem whose solutions maximize the number of satisfied constraints. In this paper, we focus on numerical CSPs, which are defined on real variables represented as floating point intervals and which constraints are numerical relations defined in intension. Solving such a problem (i.e., exactly characterizing its solution set) is generally undecidable and thus consists in providing approximations. We propose a Branch and Bound algorithm that provides under and over approximations of a solution set that maximize the maximum number \({m_{\mathcal P}}\) of satisfied constraints. The technique is applied on three numeric applications and provides promising results.     

Complexity of Approximating CSP with Balance / Hard Constraints

We study two natural extensions of Constraint Satisfaction Problems (CSPs). Balance-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of Hard-Max-CSP consists of soft constraints and hard constraints, and the goal is to maximize the weight of satisfied soft constraints while satisfying all the hard constraints. These two extensions contain many fundamental problems not captured by CSPs, and challenge traditional theories about CSPs in a more general framework. Max-2-SAT and Max-Horn-SAT are the only two nontrivial classes of Boolean CSPs that admit a robust satisfibiality algorithm, i.e., an algorithm that finds an assignment satisfying at least (1 ? g(ε)) fraction of constraints given a (1 ? ε)-satisfiable instance, where g(ε) → 0 as ε → 0, and g(0) = 0. We prove the inapproximability of these problems with balance or hard constraints, showing that each variant changes the nature of the problems significantly (in different ways). For instance, deciding whether an instance of 2-SAT admits a balanced assignment is NP-hard, and for Max-2-SAT with hard constraints, it is hard to find a constant-factor approximation even on (1 ? ε)-satisfiable instances (in particular, the version with hard constraints does not admit a robust satisfiability algorithm). We also study hardness results for a certain CSP over a larger domain capturing ordering constraints: we show that hard constraints rule out constant-factor approximation algorithms. All our hardness results are almost optimal — they completely rule out algorithms with certain properties, or can be matched by simple extensions to existing algorithms.     

Temporal Constraints: A Survey

Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about temporal occurrences and temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints represent the possible temporal relations between them. The main tasks are two: (i) deciding consistency, and (ii) answering queries about scenarios that satisfy all constraints. This paper overviews results on several classes of Temporal CSPs: qualitative interval, qualitative point, metric point, and some of their combinations. Research has progressed along three lines: (i) identifying tractable subclasses, (ii) developing exact search algorithms, and (iii) developing polynomial-time approximation algorithms. Most available techniques are based on two principles: (i) enforcing local consistency (e.g. path-consistency) and (ii) enhancing naive backtracking search.     

A new approach to partial constraint satisfaction problems

A finite binary Constraint Satisfaction Problem (CSPs) is defined as consisting of a set of n problem variables, a domain of d potential values for each variable and a set of m binary constraints involving only two variables at a time. A solution to such a CSP is specified by assignment of a value to each variable that does not violate any of the constraints. The CSPs belong to the class of NP-Complete Problems. Backtracking and its variants have been generally used for solving CSPs. The class of Partial Constraint Satisfaction Problems (PCSPs) is a subclass of CSPs that are either too difficult to solve or are unsolvable. Near optimal solutions are always desired to these problems.

In this article, we have considered only finite binary CSPs or PCSPs and developed a method of time complexity O(n ² d ²) to obtain a near optimal solution for them. The performance of the method in terms of the average number of consistency checks and the average number of constraint violations is measured on various randomly generated binary CSPs and compared with the Branch and Bound (BB) method used to obtain the same solution. The BB method is a widely used optimization technique that may be viewed as a variation of backtracking. Thus, it was a natural choice in seeking an analog of backtracking to find optimal partial solutions for PCSPs. The proposed method moves much faster to the solution. The performance results indicate that in terms of the number of consistency checks, the proposed method has much less consistency checks than BB whereas in terms of average number of constraint violations both methods are same. An upper bound on the distance of the solution from the optimal solution is obtained analytically as ?n(n???2)(d???2)/(d???1)?.     

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.     

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 unified framework for consistency check

Constraint Satisfaction Problem (CSP) involves finding values for variables to satisfy a set of constraints. Consistency check is the key technique in solving this class of problems. Past research has developed many algorithms for such a purpose, e.g., node consistency, are consistency, generalized node and arc consistency, specific methods for checking specific constraints, etc. In this article, an attempt is made to unify these algorithms into a common framework. This framework consists of two parts. the first part is a generic consistency check algorithm, which allows and encourages each individual constraint to be checked by its specific consistency methods. Such an approach provides a direct way of practical implementation of the CSP model for real problem-solving. the second part is a general schema for describing the handling of each type of constraint. the schema characterizes various issues of constraint handling in constraint satisfaction, and provides a common language for expressing, discussing, and exchanging constraint handling techniques. © 1995 John Wiley & Sons, Inc.     

Domain-dependent distributed models for railway scheduling

Many combinatorial problems can be modelled as Constraint Satisfaction Problems (CSPs). Solving a general CSP is known to be NP-complete, so closure and heuristic search are usually used. However, many problems are inherently distributed and the problem complexity can be reduced by dividing the problem into a set of subproblems. Nevertheless, general distributed techniques are not always appropriate to distribute real-life problems. In this work, we model the railway scheduling problem by means of domain-dependent distributed constraint models, and we show that these models maintained better behaviors than general distributed models based on graph partitioning. The evaluation is focused on the railway scheduling problem, where domain-dependent models carry out a problem distribution by means of trains and contiguous sets of stations.     

Accelerating filtering techniques for numeric CSPs

Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper¹ we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularities are discussed, cyclic phenomena in the propagation queue and numeric regularities of the domains of the variables. We also present in this paper an attempt to unify numeric CSPs solving methods from two distinct communities, that of CSP in artificial intelligence, and that of interval analysis.