Constraint programming is a powerful paradigm for solving combinatorial search problems that draws on a wide range of techniques from artificial intelligence, computer science, databases, programming languages, and operations research. Constraint programming is currently applied with success to many domains, such as scheduling, planning, vehicle routing, configuration, networks, and bioinformatics.

The aim of this handbook is to capture the full breadth and depth of the constraint programming field and to be encyclopedic in its scope and coverage. While there are several excellent books on constraint programming, such books necessarily focus on the main notions and techniques and cannot cover also extensions, applications, and languages. The handbook gives a reasonably complete coverage of all these lines of work, based on constraint programming, so that a reader can have a rather precise idea of the whole field and its potential. Of course each line of work is dealt with in a survey-like style, where some details may be neglected in favor of coverage. However, the extensive bibliography of each chapter will help the interested readers to find suitable sources for the missing details. Each chapter of the handbook is intended to be a self-contained survey of a topic, and is written by one or more authors who are leading researchers in the area.

The intended audience of the handbook is researchers, graduate students, higher-year undergraduates and practitioners who wish to learn about the state-of-the-art in constraint programming. No prior knowledge about the field is necessary to be able to read the chapters and gather useful knowledge. Researchers from other fields should find in this handbook an effective way to learn about constraint programming and to possibly use some of the constraint programming concepts and techniques in their work, thus providing a means for a fruitful cross-fertilization among different research areas.

The handbook is organized in two parts. The first part covers the basic foundations of constraint programming, including the history, the notion of constraint propagation, basic search methods, global constraints, tractability and computational complexity, and important issues in modeling a problem as a constraint problem. The second part covers constraint languages and solver, several useful extensions to the basic framework (such as interval constraints, structured domains, and distributed CSPs), and successful application areas for constraint programming.

- Covers the whole field of constraint programming
- Survey-style chapters
- Five chapters on applications

Table of Contents

Foreword (Ugo Montanari) 
Part I : Foundations
Chapter 1. Introduction (Francesca Rossi, Peter van Beek, Toby Walsh) 
Chapter 2. Constraint Satisfaction: An Emerging Paradigm (Eugene C. Freuder, Alan K. Mackworth) 
Chapter 3. Constraint Propagation (Christian Bessiere) 
Chapter 4. Backtracking Search Algorithms (Peter van Beek) 
Chapter 5. Local Search Methods (Holger H. Hoos, Edward Tsang) 
Chapter 6. Global Constraints (Willem-Jan van Hoeve, Irit Katriel)
Chapter 7. Tractable Structures for CSPs (Rina Dechter)
Chapter 8. The Complexity of Constraint Languages
(David Cohen, Peter Jeavons) 
Chapter 9. Soft Constraints (Pedro Meseguer, Francesca Rossi, Thomas Schiex) 
Chapter 10. Symmetry in Constraint Programming
(Ian P. Gent, Karen E. Petrie, Jean-Francois Puget)
Chapter 11. Modelling (Barbara M. Smith) 
Part II : Extensions, Languages, and Applications

Chapter 12. Constraint Logic Programming (Kim Marriott, Peter J. Stuckey, Mark Wallace) 
Chapter 13. Constraints in Procedural and Concurrent Languages (Thom Fruehwirth, Laurent Michel, Christian Schulte)
Chapter 14. Finite Domain Constraint Programming Systems (Christian Schulte, Mats Carlsson) 
Chapter 15. Operations Research Methods in Constraint Programming (John Hooker) 
Chapter 16. Continuous and Interval Constraints(Frederic Benhamou, Laurent Granvilliers) 
Chapter 17. Constraints over Structured Domains
(Carmen Gervet) 
Chapter 18. Randomness and Structure (Carla Gomes, Toby Walsh)
Chapter 19. Temporal CSPs (Manolis Koubarakis)
Chapter 20. Distributed Constraint Programming
(Boi Faltings) 
Chapter 21. Uncertainty and Change (Kenneth N. Brown, Ian Miguel)
Chapter 22. Constraint-Based Scheduling and Planning
(Philippe Baptiste, Philippe Laborie, Claude Le Pape, Wim Nuijten)
Chapter 23. Vehicle Routing (Philip Kilby, Paul Shaw) 
Chapter 24. Configuration (Ulrich Junker)
Chapter 25. Constraint Applications in Networks
(Helmut Simonis)
Chapter 26. Bioinformatics and Constraints (Rolf Backofen, David Gilbert)

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Handbook of Constraint Programming

We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

Sphere packings I

INTLAB is a toolbox for Matlab supporting real and complex intervals, and vectors, full matrices and sparse matrices over those. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available (the latter, of course, without verification of the result). Beside the basic arithmetical operations, rigorous input and output, rigorous standard functions, gradients, slopes and multiple precision arithmetic is included in INTLAB. Portability is assured by implementing all algorithms in Matlab itself with exception of exactly one routine for switching the rounding downwards, upwards and to nearest. Timing comparisons show that the used concept achieves the anticipated speed with identical code on a variety of computers, ranging from PC’s to parallel computers. INTLAB is freeware and may be copied from our home page.

INTLAB — INTerval LABoratory

The Best Worst Method (BWM) is a multi-criteria decision-making method that uses two vectors of pairwise comparisons to determine the weights of criteria. First, the best (e.g. most desirable, most important), and the worst (e.g. least desirable, least important) criteria are identified by the decision-maker, after which the best criterion is compared to the other criteria, and the other criteria to the worst criterion. A non-linear minmax model is then used to identify the weights such that the maximum absolute difference between the weight ratios and their corresponding comparisons is minimized. The minmax model may result in multiple optimal solutions. Although, in some cases, decision-makers prefer to have multiple optimal solutions, in other cases they prefer to have a unique solution. The aim of this paper is twofold: firstly, we propose using interval analysis for the case of multiple optimal solutions, in which we show how the criteria can be weighed and ranked. Secondly, we propose a linear model for BWM, which is based on the same philosophy, but yields a unique solution.

Best-worst multi-criteria decision-making method: Some properties and a linear model

INTLAB - INTerval LABoratory.

Preface to the English Edition. Preface to the German Edition. Real Interval Arithmetic. Further Concepts and Properties. Interval Evaluation and Range of Real Functions. Machine Interval Arithmetic. Complex Interval Arithmetic. Metric, Absolute, Value, and Width in. Inclusion of Zeros of a Function of One Real Variable. Methods for the Simultaneous Inclusion of Real Zeros of Polynomials. Methods for the Simultaneous Inclusion of Complex Zeros of Polynomials. Interval Matrix Operations. Fixed Point Iteration for Nonlinear Systems of Equations. Systems of Linear Equations Amenable to Interation. Optimality of the Symmetric Single Step Method with Taking Intersection after Every Component. On the Feasibility of the Gaussian Algorithm for Systems of Equations with Intervals as Coefficients. Hansen's Method. The Procedure of Kupermann and Hansen. Ireation Methods for the Inclusion of the Inverse Matrix and for Triangular Decompositions. Newton-like Methods for Nonlinear Systems of Equations. Newton-like Methods without Matrix Inversions. Newton-like Methods for Particular Systems of Nonlinear Equations. Newton-like Total step and Single Step Methods. Appendix A. The Order of Convergence of Iteration Methods in vn(Ic) and Mmn(iC) ). Appendix B. Realizations of Machine Interval Arithmetics in ALGOL 60. Appendix C. ALGOL Procedures. Bibliography. Index of Notation. Subject Index.

Jürgen Herzberger

Papers

Introduction to Interval Computation