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Parameterized Complexity of Elimination Distance to First-Order Logic Properties
arXiv - CS - Logic in Computer Science Pub Date : 2021-04-07 , DOI: arxiv-2104.02998
Fedor V. Fomin, Petr A. Golovach, Dimitrios M. Thilikos

The elimination distance to some target graph property 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 P expressible by a first order-logic formula \phi\in \Sigma_3, that is, of the form \phi=\exists x_1\exists x_2\cdots \exists x_r \forall y_1\forall y_2\cdots \forall y_s \exists z_1\exists z_2\cdots \exists z_t \psi, where \psi is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from \Sigma_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 \phi\in \Pi_3, for which computing elimination distance is W[2]-hard.

中文翻译:

消除距离到一阶逻辑属性的参数化复杂度

到某些目标图形属性P的消除距离是Bulian和Dawar引入的通用图形修改参数。我们开始研究消除距离以一阶逻辑表示的图属性。通过确定一阶逻辑公式的前缀结构上的充分必要条件,我们确定了问题的固定参数易处理性。我们的主要结果是下面的元定理:对于每个由一阶逻辑公式\ phi \ in \ Sigma_3表示的图属性P,形式为\ phi = \ exists x_1 \ exists x_2 \ cdots \ exists x_r \ forall y_1 \ forall y_2 \ cdots \ forall y_s \ exists z_1 \ exists z_2 \ cdots \ exists z_t \ psi,其中\ psi是无量词的一阶公式,用于检查图形与P的消除距离是否不超过k,是由k设定的固定参数易处理参数。可从\ Sigma_3的公式表示的图的属性包括具有界度(不包括禁止的子图)或包含有界支配集。我们通过证明这样的一般性陈述对具有甚至更具表达性的前缀结构的公式不成立,补充了该定理:存在\ Pi \ in \ Pi_3的公式,其计算消除距离为W [2] -hard。
更新日期:2021-04-08
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