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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| Published in: | Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 1 - 13 |
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| Format: | Conference Proceeding |
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29.06.2021
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Thilikos, Dimitrios M. Golovach, Petr A. Fomin, Fedor V. |
| Author_xml | – sequence: 1 givenname: Fedor V. surname: Fomin fullname: Fomin, Fedor V. email: fedor.fomin@uib.no organization: University of Bergen,Department of Informatics,Bergen,Norway – sequence: 2 givenname: Petr A. surname: Golovach fullname: Golovach, Petr A. email: petr.golovach@uib.no organization: University of Bergen,Department of Informatics,Bergen,Norway – sequence: 3 givenname: Dimitrios M. surname: Thilikos fullname: Thilikos, Dimitrios M. email: sedthilk@thilikos.info organization: Université de Montpellier,AlGCo project team, CNRS, LIRMM,Montpellier,France |
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| Snippet | The elimination distance to some target graph property {\mathcal{P}} is a general graph modification parameter introduced by Bulian and Dawar. We initiate the... |
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| SubjectTerms | Complexity theory Computer science descriptive complexity elimination distance first-order logic parameterized complexity |
| Title | Parameterized Complexity of Elimination Distance to First-Order Logic Properties |
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