Computing a partition function of a generalized pattern-based energy over a semiring

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 Gamma 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 Gamma we introduce a closure operator, overlineGammacapsupseteq Gamma, and give examples of constraint languages for which |overlineGammacap| is small. If all predicates in Gamma 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 mathcal O(|V|· |D|2 · |overlineGammacap|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 mathcal O(|V|· |overlineGammacap| · |D| · maxρ∈ Gamma|ρ|2 ) where |ρ| is the arity of ρ∈ Gamma. For a general language Gamma and non-positive weights, the minimization task can be carried out in mathcal O(|V|· |overlineGammacap|2) time. We argue that in many natural cases overlineGammacap is of moderate size, though in the worst case |overlineGammacap| can blow up and depend exponentially on maxρ∈ Gamma|ρ|.