The Descriptive Complexity of Relation Modification Problems

Florian Chudigiewitsch, Marlene Gr\"undel, Christian Komusiewicz, Nils Morawietz, Till Tantau
https://arxiv.org/abs/2603.22043 https://arxiv.org/pdf/2603.22043 https://arxiv.org/html/2603.22043

arXiv:2603.22043v1 Announce Type: new
Abstract: 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).

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