Generalizing Consistency and other Constraint Properties to Quantified Constraints

Quantified constraints and Quantified Boolean Formulae are typically much more difficult to reason with than classical constraints, because quantifier alternation makes the usual notion of solution inappropriate. As a consequence, basic properties of Constraint Satisfaction Problems (CSP), such as consistency or substitutability, are not completely understood in the quantified case. These properties are important because they are the basis of most of the reasoning methods used to solve classical (existentially quantified) constraints, and one would like to benefit from similar reasoning methods in the resolution of quantified constraints. In this paper, we show that most of the properties that are used by solvers for CSP can be generalized to quantified CSP. This requires a re-thinking of a number of basic concepts; in particular, we propose a notion of outcome that generalizes the classical notion of solution and on which all definitions are based. We propose a systematic study of the relations which hold between these properties, as well as complexity results regarding the decision of these properties. Finally, and since these problems are typically intractable, we generalize the approach used in CSP and propose weaker, easier to check notions based on locality, which allow to detect these properties incompletely but in polynomial time

XML Representation of Constraint Networks: Format XCSP 2.1

We propose a new extended format to represent constraint networks using XML. This format allows us to represent constraints defined either in extension or in intension. It also allows us to reference global constraints. Any instance of the problems CSP (Constraint Satisfaction Problem), QCSP (Quantified CSP) and WCSP (Weighted CSP) can be represented using this format

Quantified weighted constraint satisfaction problems.

Mak, Wai Keung Terrence.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 100-104).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Weighted Constraint Satisfaction Problems --- p.2Chapter 1.3 --- Quantified Constraint Satisfaction Problems --- p.3Chapter 1.4 --- Motivation and Goal --- p.4Chapter 1.5 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.7Chapter 2.1 --- Constraint Satisfaction Problems --- p.7Chapter 2.1.1 --- Backtracking Tree Search --- p.9Chapter 2.1.2 --- Local Consistencies for solving CSPs --- p.11Node Consistency (NC) --- p.13Arc Consistency (AC) --- p.14Searching by Maintaining Arc Consistency --- p.16Chapter 2.1.3 --- Constraint Optimization Problems --- p.17Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.19Chapter 2.2.1 --- Branch and Bound Search (B&B) --- p.23Chapter 2.2.2 --- Local Consistencies for WCSPs --- p.25Node Consistency --- p.26Arc Consistency --- p.28Chapter 2.3 --- Quantified Constraint Satisfaction Problems --- p.32Chapter 2.3.1 --- Backtracking Free search --- p.37Chapter 2.3.2 --- Consistencies for QCSPs --- p.38Chapter 2.3.3 --- Look Ahead for QCSPs --- p.45Chapter 3 --- Quantified Weighted CSPs --- p.48Chapter 4 --- Branch & Bound with Consistency Techniques --- p.54Chapter 4.1 --- Alpha-Beta Pruning --- p.54Chapter 4.2 --- Consistency Techniques --- p.57Chapter 4.2.1 --- Node Consistency --- p.62Overview --- p.62Lower Bound of A-Cost --- p.62Upper Bound of A-Cost --- p.66Projecting Unary Costs to Cθ --- p.67Chapter 4.2.2 --- Enforcing Algorithm for NC --- p.68Projection Phase --- p.69Pruning Phase --- p.69Time Complexity --- p.71Chapter 4.2.3 --- Arc Consistency --- p.73Overview --- p.73Lower Bound of A-Cost --- p.73Upper Bound of A-Cost --- p.75Projecting Binary Costs to Unary Constraint --- p.75Chapter 4.2.4 --- Enforcing Algorithm for AC --- p.76Projection Phase --- p.77Pruning Phase --- p.77Time complexity --- p.79Chapter 5 --- Performance Evaluation --- p.83Chapter 5.1 --- Definitions of QCOP/QCOP+ --- p.83Chapter 5.2 --- Transforming QWCSPs into QCOPs --- p.90Chapter 5.3 --- Empirical Evaluation --- p.91Chapter 5.3.1 --- Random Generated Problems --- p.92Chapter 5.3.2 --- Graph Coloring Game --- p.92Chapter 5.3.3 --- Min-Max Resource Allocation Problem --- p.93Chapter 5.3.4 --- Value Ordering Heuristics --- p.94Chapter 6 --- Concluding Remarks --- p.96Chapter 6.1 --- Contributions --- p.96Chapter 6.2 --- Limitations and Related Works --- p.97Chapter 6.3 --- Future Works --- p.99Bibliography --- p.10

QCSP made practical by virtue of restricted quantification

The QCSP + language we introduce extends the framework of Quantified Constraint Satisfaction Problems (QCSPs) by enabling us to neatly express restricted quantifications via a chain of nested CSPs to be interpreted as alternately conjuncted and disjuncted. Restricted quantifiers turn out to be a convenient solution to the crippling modeling issues we encounter in QCSP and—surprisingly— they help to reuse propagation technology and to prune the search space. Our QCSP + solver—which also handles arithmetic and global constraints— exhibits state-of-the-art performances.