Computing a partition function of a generalized pattern-based energy over a semiring
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Valued constraint satisfaction problems with ordered variables (VCSPO) are a special case of Valued CSPs in which variables are totally ordered and soft constraints are imposed on tuples of variables that do not violate the order. We study a restriction of VCSPO, in which soft constraints are imposed on a segment of adjacent variables and a constraint language Γ consists of {0,1}-valued characteristic functions of predicates. This kind of potentials generalizes the so-called pattern-based potentials, which were applied in many tasks of structured prediction. For a constraint language Γ we introduce a closure operator, Γ^∩⊇Γ, and give examples of constraint languages for which |Γ^∩| is small. If all predicates in Γ are cartesian products, we show that the minimization of a generalized pattern-based potential (or, the computation of its partition function) can be made in 𝒪(|V|· |D|^2 · |Γ^∩|^2 ) time, where V is a set of variables, D is a domain set. If, additionally, only non-positive weights of constraints are allowed, the complexity of the minimization task drops to 𝒪(|V|· |Γ^∩| · |D| ·max_ρ∈Γρ^2 ) where ρ is the arity of ρ∈Γ. For a general language Γ and non-positive weights, the minimization task can be carried out in 𝒪(|V|· |Γ^∩|^2) time. We argue that in many natural cases Γ^∩ is of moderate size, though in the worst case |Γ^∩| can blow up and depend exponentially on max_ρ∈Γρ.
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