We investigate the fine-grained and the parameterized complexity of several generalizations of binary constraint satisfaction problems (BINARY-CSPs), that subsume variants of graph colouring problems. Our starting point is the observation that several algorithmic approaches that resulted in complexity upper bounds for these problems, share a common structure. We thus explore an algebraic approach relying on semirings that unifies different generalizations of BINARY-CSPs (such as the counting, the list, and the weighted versions), and that facilitates a general algorithmic approach to efficiently solving them. The latter is inspired by the (component) twin-width parameter introduced by Bonnet et al., which we generalize via edge-labelled graphs in order to formulate it to arbitrary binary constraints. We consider input instances with bounded component twin-width, as well as constraint templates of bounded component twin-width, and obtain an FPT algorithm as well as an improved, exponential-time algorithm, for broad classes of binary constraints. We illustrate the advantages of this framework by instantiating our general algorithmic approach on several classes of problems (e.g., the $H$-coloring problem and its variants), and showing that it improves the best complexity upper bounds in the literature for several well-known problems.
翻译:我们研究了二进制约束满意度问题(BINARY-CSPs)的若干一般性问题(BINARY-CSPs)的细微复杂度和参数复杂度,它包含了图形颜色问题的变异性。我们的出发点是观察到导致这些问题的复杂上界的几种算法方法具有共同的结构。我们因此探索了一种代数法方法,它依靠的是将二进制约束满意度(BINARY-CSPs)的不同一般化(例如计数、列表和加权版本)的半灵精度法,并且有助于采用一般算法方法来有效解决这些问题。后者的灵感来自Bonnet et et 等人 推出的(构件) 双维参数,我们通过边缘标签图将其加以概括,以便形成任意的二进制限制。我们考虑的是带有两进制组件的结合性输入实例,以及捆绑的两维成组件(例如计算、列表和加权版本)的制约性模板,并获得一种已知的、指数-时间算法,用以有效解决这些问题。我们举例说明了这个框架的优点,就是由BON等人等人等人等人等人等引入的双进制的双进制的双进制法方法,从而在几进制问题上展示。我们一般的复杂度和变式中呈现了几进制问题。