Inspired by recent results for the so-called fair and local linear problems, we define two problems, Local Linear Set and Local Linear Vertex Cover. Here the task is to find a set of vertices $X\subseteq V$ (which is a vertex cover) of the input graph $(V,E)$ such that the local linear constraints $\ell (v)\le |X\cap N(v)|\le u(v)$ hold for all $v\in V$; here, $\ell (v)$ and $u(v)$ are on the input. We prove that both problems are in FPT for the parameter twin cover number (using integer linear programming).

中文翻译：

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具有有界孪生覆盖数的图上的局部线性集

受最近所谓的公平和局部线性问题的结果启发，我们定义了两个问题，即局部线性集和局部线性顶点覆盖。这里的任务是找到一组顶点$X\subseteq \mathrm{伏特}$ 输入图的（是一个顶点覆盖） $（\mathrm{伏特}，E）$ 这样局部线性约束 $\ell （v）\le |X\cap ñ（v）|\le ü（v）$ 坚持所有人 $v\in \mathrm{伏特}$; 这里，$\ell （v）$ 和 $ü（v）$在输入上。我们证明两个问题都在FPT参数双覆盖号（使用整数线性编程）中。