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).

Figures (6)

• Figure 1: Visualization of the reduction in Lemma \ref{['lemma:p-undir-ae']}.
• Figure 2: Visualization of the reduction in Lemma \ref{['lemma:p-basic-eae']}.
• Figure 3: Example for the reduction in Lemma \ref{['lemma:basic-eaa']}.
• Figure 4: Visualization of the reduction in Lemma \ref{['lemma:p-basic-aea']}.
• Figure 5: Visualization of the reduction in Lemma \ref{['lemma:p-basic-aee']}.

Theorems & Definitions (53)

• theorem 1
• lemma 1
• proof
• lemma 2
• proof
• theorem 2
• lemma 3
• proof
• lemma 4: $\star$
• lemma 5: $\star$