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This book presents general methods for analysing the complexity of optimisation problems cast as valued constraint satisfaction problems (VCSPs). It is an ideal resource for researchers in constraint programming and discrete optimisation.

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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.

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Chap. 1 Introduction.- Chap. 2 Background.- Chap. 3 Expressibility of Valued Constraints.- Chap. 4 Expressibility of Fixed-Arity Languages.- Chap. 5 Expressibility of Submodular Languages.- Chap. 6 Non-expressibility of Submodular Languages.- Chap. 7 Tractable Languages.- Chap. 8 Conservative Languages.- Chap. 9 The Power of Linear Programming.- Chap. 10 Hybrid Tractability.- Chap. 11 Summary and Open Problems.- References.- Index.

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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)

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