© 2012

The Complexity of Valued Constraint Satisfaction Problems


Part of the Cognitive Technologies book series (COGTECH)

Table of contents

  1. Front Matter
    Pages I-XVII
  2. Background

    1. Front Matter
      Pages 1-1
    2. Stanislav Živný
      Pages 3-27
  3. Expressive Power

    1. Front Matter
      Pages 29-29
    2. Stanislav Živný
      Pages 31-47
    3. Stanislav Živný
      Pages 49-78
    4. Stanislav Živný
      Pages 79-94
    5. Stanislav Živný
      Pages 95-114
  4. Tractability

    1. Front Matter
      Pages 115-115
    2. Stanislav Živný
      Pages 117-124
    3. Stanislav Živný
      Pages 125-129
    4. Stanislav Živný
      Pages 131-135
    5. Stanislav Živný
      Pages 137-149
    6. Stanislav Živný
      Pages 151-152
  5. Back Matter
    Pages 153-170

About this book


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)

Authors and affiliations

  1. 1.University CollegeUniversity of OxfordOxfordUnited Kingdom

About the authors

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.

Bibliographic information

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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)