The Descriptive Complexity of Relation Modification Problems
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).
Cite
@article{arxiv.2603.22043,
title = {The Descriptive Complexity of Relation Modification Problems},
author = {Florian Chudigiewitsch and Marlene Gründel and Christian Komusiewicz and Nils Morawietz and Till Tantau},
journal= {arXiv preprint arXiv:2603.22043},
year = {2026}
}