Xuan-Ha Vu

Using directed acyclic graphs to coordinate propagation and search for numerical constraint satisfaction problems

The paper of H. Schichl & A. Neumaier has given the fundamentals of interval analysis on DAGs for global optimization and constraint propagation. We show in This work how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm, which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties.

Numerical Constraint Satisfaction Problems with Non-isolated Solutions

In recent years, interval constraint-based solvers have shown their ability to efficiently solve complex instances of non-linear numerical CSPs. However, most of the working systems are designed to deliver point-wise solutions with an arbitrary accuracy. This works generally well for systems with isolated solutions but less well when there is a continuum of feasible points (e.g. under-constrained problems, problems with inequalities). In many practical applications, such large sets of solutions express equally relevant alternatives which need to be identified as completely as possible. In this paper, we address the issue of constructing concise inner and outer approximations of the complete solution set for non-linear CSPs. We propose a technique which combines the extreme vertex representation of orthogonal polyhedra 1,2,3, as defined in computational geometry, with adapted splitting strategies 4 to construct the approximations as unions of interval boxes. This allows for compacting the explicit representation of the complete solution set and improves efficiency.

Approximation Techniques for Non-linear Problems with Continuum of Solutions

Most of the working solvers for numerical constraint satisfaction problems (NCSPs) are designed to delivering point-wise solutions with an arbitrary accuracy. When there is a continuum of feasible points this might lead to prohibitively verbose representations of the output. In many practical applications, such large sets of solutions express equally relevant alternatives which need to be identified as completely as possible. The goal of this paper is to show that by using appropriate approximation techniques, explicit representations of the solution sets, preserving both accuracy and completeness, can still be proposed for NCSPs with continuum of solutions. We present a technique for constructing concise inner and outer approximations as unions of interval boxes. The proposed technique combines a new splitting strategy with the extreme vertex representation of orthogonal polyhedra [1,2,3], as defined in computational geometry. This allows for compacting the representation of the approximations and improves efficiency.

Enhancing numerical constraint propagation using multiple inclusion representations

Building tight and conservative enclosures of the solution set is of crucial importance in the design of efficient complete solvers for numerical constraint satisfaction problems (NCSPs). This paper proposes a novel generic algorithm enabling the cooperative use, during constraint propagation, of multiple enclosure techniques. The new algorithm brings into the constraint propagation framework the strength of techniques coming from different areas such as interval arithmetic, affine arithmetic, and mathematical programming. It is based on the directed acyclic graph (DAG) representation of NCSPs whose flexibility and expressiveness facilitates the design of fine-grained combination strategies for general factorable systems. The paper presents several possible combination strategies for creating practical instances of the generic algorithm. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming illustrate the flexibility and efficiency of the approach.

Combining multiple inclusion representations in numerical constraint propagation

This work proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from different areas. The scheme is based on the DAG representation of the constraint system. This enables devising fine-grained combination strategies involving any factorable constraint system. The paper presents several possible combination strategies for creating practical instances of the generic scheme. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic and linear programming illustrate the flexibility and efficiency of the approach.

Clustering for Disconnected Solution Sets of Numerical CSPs

This paper considers the issue of postprocessing the output of interval-based solvers for further exploitations when solving numerical CSPs with continuum of solutions. Most interval-based solvers cover the solution sets of such problems with a large collection of boxes. This makes it difficult to exploit their results for other purposes than simple querying. For many practical problems, it is highly desirable to run more complex queries on the representations of the solution set. We propose to use clustering techniques to regroup the output in order to provide some characteristics of the solution set. Three new clustering algorithms based on connectedness and their combinations are proposed.