The Complexity of Valued Constraint Satisfaction Problems

1. Front Matter

   Pages I-XVII

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2. Background

   1. Front Matter

      Pages 1-1

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3. Part I

   1. Background

      • Stanislav Živný

      Pages 3-27

4. Expressive Power

   1. Front Matter

      Pages 29-29

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   2. Expressibility of Valued Constraints

      • Stanislav Živný

      Pages 31-47

   3. Expressibility of Fixed-Arity Languages

      • Stanislav Živný

      Pages 49-78

   4. Expressibility of Submodular Languages

      • Stanislav Živný

      Pages 79-94

   5. Non-expressibility of Submodular Languages

      • Stanislav Živný

      Pages 95-114

5. Tractability

   1. Front Matter

      Pages 115-115

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   2. Tractable Languages

      • Stanislav Živný

      Pages 117-124

   3. Conservative Languages

      • Stanislav Živný

      Pages 125-129

   4. The Power of Linear Programming

      • Stanislav Živný

      Pages 131-135

   5. Hybrid Tractability

      • Stanislav Živný

      Pages 137-149

   6. Summary and Open Problems

      • Stanislav Živný

      Pages 151-152

6. Back Matter

   Pages 153-170

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The topic of this book is the following optimisation problem: given a set of discrete variables and a set of functions, each depending on a subset of the variables, minimise the sum of the functions over all variables. This fundamental research problem has been studied within several different contexts of discrete mathematics, computer science and artificial intelligence under different names: Min-Sum problems, MAP inference in Markov random fields (MRFs) and conditional random fields (CRFs), Gibbs energy minimisation, valued constraint satisfaction problems (VCSPs), and, for two-state variables, pseudo-Boolean optimisation.

In this book the author presents general techniques for analysing the structure of such functions and the computational complexity of the minimisation problem, and he gives a comprehensive list of tractable cases. Moreover, he demonstrates that the so-called algebraic approach to VCSPs can be used not only for the search for tractable VCSPs, but also for other questions such as finding the boundaries to the applicability of certain algorithmic techniques.

The book is suitable for researchers interested in methods and results from the area of constraint programming and discrete optimisation.

• Algorithms
• Constraint programming (CP)
• Constraint satisfaction problem (CSP)
• Discrete optimization
• Linear programming (LP)
• Tractability
• Tractable languages
• Valued Constraint satisfaction problems (VCSPs)

From the reviews:

“The book is concerned with valued constraint satisfaction problems with finite domains, i. e., the problem of minimising the sum of functions that each depend on some subset of a finite set of discrete variables with finite domains. … In all places the description is very precise and exact, detailed where needed. This makes it a very useful reference for researchers working in the field.” (Thomas Jansen, zbMATH, Vol. 1283, 2014)

• University College, University of Oxford, Oxford, United Kingdom

  Stanislav Živný

Dr. Stanislav Živný has a Ph.D. in Computer Science from the University of Oxford, and he was a Junior Research Fellow in Mathematical and Physical Sciences at Oxford's University College. The doctoral thesis won the 2011 ACP (Association for Constraint Programming) Doctoral Award for the best thesis in the area of constraint programming from the previous two years. He was awarded a Senior Research Fellowship in Advances in Discrete Mathematics and its Applications, hosted by the University of Warwick, which commenced in autumn 2012.