A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the classical and parameterized complexity of relation modification problems - the latter w. r. t. the modification budget k - based on the descriptive complexity of the respective target property. We consider different types of logical structures on which modifications are performed: Whereas monadic structures and undirected graphs without self-loops each yield their own complexity landscapes, we find that modifying undirected graphs with self-loops, directed graphs, or arbitrary logical structures is equally hard w. r. t. quantifier patterns. Moreover, we observe that all classes of problems considered in this paper are subject to a strong dichotomy in the sense that they are either very easy to solve (that is, they lie in paraAC^{0\uparrow} or TC^0) or intractable (that is, they contain W[2]-hard or NP-hard problems).

翻译：关系修改问题以逻辑结构和自然数k为输入，询问是否可以通过对结构进行k次修改使其满足预定义性质。我们基于目标性质的描述复杂度，对关系修改问题的经典复杂度与参数化复杂度（后者关于修改预算k）进行了完整分类。我们考虑执行修改的不同类型逻辑结构：尽管一元结构和无自环无向图各自产生不同的复杂度景观，但我们发现，在量词模式方面，修改带自环无向图、有向图或任意逻辑结构具有相同难度。此外，我们观察到本文所考虑的所有问题类别均服从强二分性，即它们要么非常容易求解（属于 paraAC^{0\uparrow} 或 TC^0），要么难以处理（包含 W[2]-困难或 NP-困难问题）。