Background -- Expressive power -- Expressibility of valued constraints -- Expressibility of fixed-Arity languages -- Expressibility of submodular languages -- Non-expressibility of submodular languages -- Tractability -- Tractable languages -- Conservative languages -- The power of linear programming -- Hybrid tractability -- Summary and open problems
Summary
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 variabl