Constraint Consistency Techniques for Continuous Domains

Ing enieure informaticienne EPFL originaire de Plaaein (FR) accet ee sur proposition du jury: Abstract Constraint Satisfaction Problems (CSPs) are ubiquitous in computer science. Many problems , ranging from resource allocation and scheduling to fault diagnosis and design, involve constraint satisfaction as an essential component. A CSP is given by a set of variables and constraints on small subsets of these variables. It is solved by nding assignments of values to the variables such that all constraints are satissed. In its most general form, a CSP is combinatorial and complex. In this thesis, we consider constraint satisfaction problems with variables in continuous, numerical domains. Contrary to most existing techniques, which focus on computing a single optimal solution, we address the problem of computing a compact representation of the space of all solutions that satisfy the constraints. This has the advantage that no optimization criterion has to be formulated beforehand, and that the space of possibilities can be explored systematically. In certain applications, such as diagnosis and design, these advantages are crucial. In consistency techniques, the solution space is represented by labels assigned to individual variables or combinations of variables. When the labeling is globally consistent, each label contains only those values or combinations of values which appear in at least one solution. This kind of labeling is a compact, sound and complete representation of the solution space, and can be combined with other reasoning methods. In practice, computing a globally consistent labeling is too complex. This is usually tackled in two ways. One way is to enforce consistencies locally, using propagation algorithms. This prunes the search space and hence reduces the subsequent search eeort. The other way is to identify simplifying properties which guarantee that global consistency can be enforced tractably using local propagation algorithms. When constraints are represented by mathematical expressions, implementing local consistency algorithms is diicult because it requires tools for solving arbitrary systems of equations. In this thesis, we propose to approximate feasible solution regions by 2 k-trees, thus providing a means of combining constraints logically rather than numerically. This representation, commonly used in computer vision and image processing, avoids using complex mathematical tools. We propose simple and stable algorithms for computing labels of arbitrary degrees of consistency using this representation. For binary constraints, it is known that simplifying convexity properties reduces the complexity of solving a CSP. These properties guarantee that local degrees of consistency are …