发表机构
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究分类了零强制与电力支配问题变体的参数化复杂性，发现初始化、轮数限制和强制阈值的微小变化会导致从FPT到W[P]-完全的复杂性剧变。
AI 中文摘要
零强制（Zero Forcing, ZF）和电力支配集（Power Dominating Set, PDS）基于从一组特定于问题的初始标记顶点开始的共同强制过程来标记图中的顶点。ZF初始标记所选顶点，而PDS还额外标记这些顶点的邻居。在强制过程中，一个仅有一个未标记邻居的已标记顶点可以强制该邻居，使其也变为已标记。通过穷举应用此规则，一个解即可标记整个图。一种变体推广了强制阈值；当顶点具有固定数量$k$个未标记邻居时，它们可以进行强制。另一种变体将传播限制为固定轮数。我们分类了通过组合这些初始化、轮数限制和强制阈值的选择所获得的问题变体的参数化复杂性。我们表明，通过适当的选择，这些变体在参数化复杂性上范围从固定参数可处理到对$W$-层次结构的每一偶数层$W[2\ell]$完全，并直至$W[P]$-完全。我们的结果表明，这三个维度中任何一个的微小变化都可能导致问题复杂性的急剧变化。
英文摘要
Zero Forcing (ZF) and Power Dominating Set (PDS) mark vertices in a graph based on a common forcing process starting from a problem-specific set of initially marked vertices. ZF initially marks the selected vertices while PDS additionally marks their neighbors. In the forcing process, a marked vertex with only one unmarked neighbor may force that neighbor which then becomes marked, too. A solution marks the entire graph by exhaustive application of this rule. One variant generalizes the forcing threshold; vertices may force when they have a fixed number of $k$ unmarked neighbors. Another variant limits propagation to a fixed number of rounds. We classify the parameterized complexity of the problem variants obtained by combining these choices of initialization, round limit and forcing threshold. We show that with appropriate choices, these variants range in parameterized complexity from fixed-parameter tractable to complete for every even layer $W[2\ell]$ of the $W$-hierarchy, and up to $W[P]$-complete. Our results demonstrate that small changes in any one of these three dimensions can lead to a sharp change in problem complexity.
