永恒顶点覆盖的结构参数化
Structural Parameterizations for Eternal Vertex Cover
- Indian Institute of Technology, Gandhinagar（印度理工学院甘地纳加尔分校）
- School of Computer Science, University of Leeds（利兹大学计算机科学学院）
- Sapienza University of Rome（罗马第一大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究永恒顶点覆盖问题的结构参数化，证明其关于簇顶点删除数为FPT，关于顶点完整性为XP，并给出加性近似算法及针对特殊连通分量的FPT算法。
AI中文摘要：
永恒顶点覆盖（Eternal Vertex Cover, EVC）是一种在无向图 $G$ 上进行的回合制攻防博弈。游戏开始时，防守方在 $G$ 的顶点上放置 $k$ 个守卫。攻击方在其回合可以选择一条两个端点均未被守卫占据的边 $e$ 进行“攻击”。如果有一个守卫沿着边 $e$ 移动，则该边被防守。防守方在其回合可以移动任意子集的守卫，每个守卫只能移动到相邻顶点。为了无限期地防御任意攻击序列所需的最少守卫数称为永恒顶点覆盖数，它推广了经典的顶点覆盖数。确定该数在一般情况下是 NP 困难的，这促使了对参数化算法和近似算法的研究。已知当以覆盖数为参数时，该问题是 FPT 的，但结构参数在文献中相对未被探索。在这项工作中，我们探索了 EVC 的结构参数化。我们证明了 EVC 在以簇顶点删除数为参数时是 FPT 的，这推广了先前以顶点覆盖数为参数的研究。接下来，我们研究以顶点完整性为参数的问题，顶点完整性是指需要从 $G$ 中删除的最少顶点数，使得结果图是不相交的常数大小连通分量的并集。我们首先证明永恒顶点覆盖在以顶点完整性为参数时是 XP 的。然后，我们开发了一个多项式时间近似算法，该算法计算一个加性 $6k+1$（$g(k)$）近似，其中 $k$ 等于簇顶点删除数（顶点完整性）。最后，我们展示了一个 FPT 算法，适用于删除集产生“良好”连通分量的情况，这些分量的大小有界且满足一个技术条件。
英文摘要：
Eternal Vertex Cover (EVC) is a turn-based attacker-defender game on an undirected graph $G$. To begin with, the defender places $k$ guards on vertices of $G$. The attacker, on their turn, can choose an edge $e$ not already occupied at both endpoints to "attack". The edge $e$ is defended if a guard moves along the edge $e$. The defender, on their turn, can move any subset of guards. A guard can only move to a neighboring vertex. The minimum number of guards needed to indefinitely defend against any sequence of attacks is called the eternal vertex cover number, generalizing the classic vertex cover number. Determining this number is NP-hard in general, motivating the study of parameterized and approximation algorithms. The problem is known to be FPT when parameterized by the cover number, but structural parameters remain relatively unexplored in the literature. In this work, we explore structural parameterizations for EVC. We show that EVC is FPT parameterized by the cluster vertex deletion number, which generalizes the previously studied parameterization by vertex cover number. We next study the problem parameterized by vertex integrity, which is the smallest number of vertices we need to delete from $G$ so that the resulting graph is a disjoint union of constant-sized components. We first show that Eternal Vertex Cover is XP parameterized by vertex integrity. Then, we develop a polynomial-time approximation algorithm, which computes an additive $6k+1$ ($g(k)$) approximation, where $k$ is equal to the cluster vertex deletion number (vertex integrity). Finally, we show a FPT algorithm for when the deletion set produces "nice" connected components, which are components that are bounded in size and satisfy a technical condition.
