Parameterized Complexity of Elimination Distance to First-Order Logic Properties

The elimination distance to some target graph property {\mathcal{P}} is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractabili...

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Bibliographic Details
Published in:Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 1 - 13
Main Authors: Fomin, Fedor V., Golovach, Petr A., Thilikos, Dimitrios M.
Format: Conference Proceeding
Language:English
Published: IEEE 29.06.2021
Subjects:
Complexity theory
Computer science
descriptive complexity
elimination distance
first-order logic
parameterized complexity
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Summary:The elimination distance to some target graph property {\mathcal{P}} is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: For every graph property {\mathcal{P}} expressible by a first order-logic formula φ ∈ Σ 3 , that is, of the form\begin{equation*}\varphi = \exists {x_1}\exists {x_2} \cdots \exists {x_r}\forall {y_1}\forall {y_2} \cdots \forall {y_s}\quad \exists {z_1}\exists {z_2} \cdots \exists {z_t}\,\psi ,\end{equation*}where ψ is a quantifier-free first-order formula, checking whether the elimination distance of a graph to {\mathcal{P}} does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Σ 3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: There are formulas φ ∈ Π 3 , for which computing elimination distance is W[2]-hard.
DOI:10.1109/LICS52264.2021.9470540