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

Abstract

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

Tasks

Reproductions