We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range \(\delta >0\). In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most \(\delta \) from one of these facilities. We investigate this covering problem in terms of the rational parameter \(\delta \). We prove that the problem is polynomially solvable whenever \(\delta \) is a unit fraction, and that the problem is NP-hard for all non unit fractions \(\delta \). We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for \(\delta <3/2\), and it is W[2]-hard for \(\delta \ge 3/2\).

我们研究了无向图上的连续设施位置问题，其中所有边都有单位长度，设施可能位于顶点以及边的内部点上。目标是用最少数量的设施覆盖整个图，覆盖范围\(\delta >0\)。换句话说，我们希望尽可能少地放置设施，条件是每条边上的每个点与这些设施之一的距离最多为\(\delta \)。我们根据有理参数\(\delta \)来研究这个覆盖问题。我们证明，只要\(\delta \)是单位分数，该问题是多项式可解的，并且对于所有非单位分数，该问题都是 NP-hard\(\delta \)。我们还分析了以解大小为参数的参数化复杂度：对于\(\delta <3/2\) ，得到的问题是固定参数易处理的，对于\(\delta \ge 3/，它是W [2]-hard 2\)。