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 \(\varvec{\Gamma }\) consists of \({\textbf {\{0,1\}}}\)-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 \(\varvec{\Gamma }\) we introduce a closure operator, \( \overline{\varvec{\Gamma }^{\cap }}\supseteq \varvec{\Gamma }\), and give examples of constraint languages for which \(|\overline{\varvec{\Gamma }^{\cap }}|\) is small. If all predicates in \(\varvec{\Gamma }\) 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 \({\varvec{\mathcal O}}(|\varvec{V}|\cdot |\varvec{D}|\varvec{^2} \cdot |\overline{\varvec{\Gamma }^{\cap }}|\varvec{^2})\) time, where \(\varvec{V}\) is a set of variables, \(\varvec{D}\) is a domain set. If, additionally, only non-positive weights of constraints are allowed, the complexity of the minimization task drops to \({\varvec{\mathcal O}}(|\varvec{V}|\cdot |\overline{\varvec{\Gamma }^{\cap }}| \cdot |\varvec{D}| \cdot \varvec{\max }_{\varrho \in \varvec{\Gamma }}\Vert \varrho \Vert \varvec{^2})\) where \(\Vert \varrho \Vert \) is the arity of \(\varrho \in \varvec{\Gamma }\). For a general language \(\varvec{\Gamma }\) and non-positive weights, the minimization task can be carried out in \({\varvec{\mathcal O}}(|\varvec{V}|\cdot |\overline{\varvec{\Gamma }^{\cap }}|\varvec{^2})\) time. We argue that in many natural cases \(\overline{\varvec{\Gamma }^{\cap }}\) is of moderate size, though in the worst case \(|\overline{\varvec{\Gamma }^{\cap }}|\) can blow up and depend exponentially on \(\varvec{\max }_{\varrho \in \varvec{\Gamma }}\Vert \varrho \Vert \).

有序变量的值化约束满足问题 (VCSPO) 是值化 CSP 的一种特殊情况，其中变量是完全有序的，并且对不违反顺序的变量元组施加软约束。我们研究了 VCSPO 的限制，其中对一段相邻变量施加软约束，约束语言\(\varvec{\Gamma }\)由\({\textbf {\{0,1\}}}组成\) -谓词的定值特征函数。这种势概括了所谓的基于模式的势，它被应用于结构化预测的许多任务中。对于约束语言\(\varvec{\Gamma }\)，我们引入一个闭包运算符\( \overline{\varvec{\Gamma }^{\cap }}\supseteq \varvec{\Gamma }\)，并给出\(|\overline{\varvec{\Gamma }^{\cap }}|\)较小的约束语言的示例。如果\(\varvec{\Gamma }\)中的所有谓词都是笛卡尔积，我们表明可以在\({\varvec{ \mathcal O}}(|\varvec{V}|\cdot |\varvec{D}|\varvec{^2} \cdot |\overline{\varvec{\Gamma }^{\cap }}|\varvec{ ^2})\)时间，其中\(\varvec{V}\)是变量集，\(\varvec{D}\)是域集。此外，如果仅允许非正约束权重，则最小化任务的复杂性降至\({\varvec{\mathcal O}}(|\varvec{V}|\cdot |\overline{\varvec{\Gamma }^{\cap }}| \cdot |\varvec{D}| \cdot \ varvec{\max }_{\varrho \in \varvec{\Gamma }}\Vert \varrho \Vert \varvec{^2})\) 其中 \(\Vert \varrho \Vert \ )是\ ( \varrho \in \varvec{\Gamma }\)。对于通用语言\(\varvec{\Gamma }\)和非正权重，最小化任务可以在\({\varvec{\mathcal O}}(|\varvec{V}|\cdot | \overline{\varvec{\Gamma }^{\cap }}|\varvec{^2})\)时间。我们认为在许多自然情况下\(\overline{\varvec{\Gamma }^{\cap }}\) 的大小适中，但在最坏的情况下\(|\overline{\varvec{\Gamma }^{ \帽}}|\)可以爆炸并呈指数依赖于\(\varvec{\max }_{\varrho \in \varvec{\Gamma }}\Vert \varrho \Vert \)。