The Descriptive Complexity of Relation Modification Problems
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917358125187072 |
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| author | Chudigiewitsch, Florian Gründel, Marlene Komusiewicz, Christian Morawietz, Nils Tantau, Till |
| author_facet | Chudigiewitsch, Florian Gründel, Marlene Komusiewicz, Christian Morawietz, Nils Tantau, Till |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22043 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Descriptive Complexity of Relation Modification Problems Chudigiewitsch, Florian Gründel, Marlene Komusiewicz, Christian Morawietz, Nils Tantau, Till Logic in Computer Science Computational Complexity 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). |
| title | The Descriptive Complexity of Relation Modification Problems |
| topic | Logic in Computer Science Computational Complexity |
| url | https://arxiv.org/abs/2603.22043 |