The Descriptive Complexity of Relation Modification Problems

TL;DR

This paper classifies the computational complexity of relation modification problems based on the logical properties of the structures involved, revealing a dichotomy between easy and hard cases.

Contribution

It provides a complete classification of the classical and parameterized complexity of relation modification problems based on descriptive logic.

Findings

Different logical structures have distinct complexity landscapes.

Modifying undirected graphs with self-loops, directed graphs, or arbitrary structures is equally hard.

Problems are either very easy (in paraAC^0 or TC^0) or intractable (W[2]-hard or NP-hard).

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).