The Descriptive Complexity of Relation Modification Problems

2026-03-23

AI Summary

This study investigates the decision problem of whether a given logical property can be satisfied by a relational structure—such as an undirected or directed graph—after at most k modifications to its relations. Employing tools from descriptive complexity theory and parameterized complexity analysis, the work systematically characterizes the relationship between the logical expressiveness of the target property—particularly its quantifier structure—and the computational difficulty of the associated modification problem. The main contribution lies in establishing a complete complexity classification for this problem across various structural settings, revealing a striking dichotomy: every instance is either of very low complexity (belonging to para-AC⁰↑ or TC⁰) or is W[2]-hard or NP-hard. This result delineates a unified boundary of computational hardness grounded in logical definability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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).

Problem

• relation modification
• descriptive complexity
• parameterized complexity
• logical structures
• complexity classification

Innovation

• descriptive complexity
• relation modification
• parameterized complexity
• dichotomy
• logical structures