We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance \(\delta\) from each other. We investigate the complexity of this problem in terms of the rational parameter \(\delta\). The problem is polynomially solvable, if the numerator of \(\delta\) is 1 or 2, while all other cases turn out to be NP-hard.

中文翻译：

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在图上分散令人讨厌的设施

我们在图形上研究了一个连续的设施位置问题，其中所有边缘都有单位长度，并且设施也可能位于边缘内部。目标是在任何两个设施彼此之间至少具有距离\（\ delta \）的条件下，定位尽可能多的设施。我们根据有理参数\（\ delta \）来研究此问题的复杂性。如果\（\ delta \）的分子是1或2，则该问题可以通过多项式解决，而其他所有情况都证明是NP-hard的。