Boolean satisfiability is probably the most studied of the combinatorial optimization/search problems. Significant effort has been devoted to trying to provide practical solutions to this problem for problem instances encountered in a range of applications in electronic design automation (EDA), as well as in artificial intelligence (AI). This study has culminated in the development of several SAT packages, both proprietary and in the public domain (e.g. GRASP, SATO) which find significant use in both research and industry. Most existing complete solvers are variants of the Davis-Putnam (DP) search algorithm. In this paper we describe the development of a new complete solver, Chaff which achieves significant performance gains through careful engineering of all aspects of the search-especially a particularly efficient implementation of Boolean constraint propagation (BCP) and a novel low overhead decision strategy. Chaff has been able to obtain one to two orders of magnitude performance improvement on difficult SAT benchmarks in comparison with other solvers (DP or otherwise), including GRASP and SATO.

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Chaff: engineering an efficient SAT solver

We describe an approximation method for solving the minimum makespan problem of job shop scheduling. It sequences the machines one by one, successively, taking each time the machine identified as a bottleneck among the machines not yet sequenced. Every time after a new machine is sequenced, all previously established sequences are locally reoptimized. Both the bottleneck identification and the local reoptimization procedures are based on repeatedly solving certain one-machine scheduling problems. Besides this straight version of the Shifting Bottleneck Procedure, we have also implemented a version that applies the procedure to the nodes of a partial search tree. Computational testing shows that our approach yields consistently better results than other procedures discussed in the literature. A high point of our computational testing occurred when the enumerative version of the Shifting Bottleneck Procedure found in a little over five minutes an optimal schedule to a notorious ten machines/ten jobs problem on which many algorithms have been run for hours without finding an optimal solution.

The Shifting Bottleneck Procedure for Job Shop Scheduling

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 an approximation algorithm for the problem of finding the minimum makespan in a job shop. The algorithm is based on simulated annealing, a generalization of the well known iterative improvement approach to combinatorial optimization problems. The generalization involves the acceptance of cost-increasing transitions with a nonzero probability to avoid getting stuck in local minima. We prove that our algorithm asymptotically converges in probability to a globally minimal solution, despite the fact that the Markov chains generated by the algorithm are generally not irreducible. Computational experiments show that our algorithm can find shorter makespans than two recent approximation approaches that are more tailored to the job shop scheduling problem. This is, however, at the cost of large running times.

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Job Shop Scheduling by Simulated Annealing

A fast and easily implementable approximation algorithm for the problem of finding a minimum makespan in a job shop is presented. The algorithm is based on a taboo search technique with a specific neighborhood definition which employs a critical path and blocks of operations notions. Computational experiments up to 2,000 operations show that the algorithm not only finds shorter makespans than the best approximation approaches but also runs in shorter time. It solves the well-known 10 × 10 hard benchmark problem within 30 seconds on a personal computer.

A Fast Taboo Search Algorithm for the Job Shop Problem

(1982). Sequencing and Scheduling: An Introduction to the Mathematics of the Job-Shop. Journal of the Operational Research Society: Vol. 33, No. 9, pp. 862-862.

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Graham K. Rand

Papers

Sequencing and Scheduling: An Introduction to the Mathematics of the Job-Shop