AI 中文总结
研究$δ$-分散这一连续设施选址问题的可近似性，根据$δ$值不同证明了多APX难或APX难，借助平移定理将重点置于特定区间，给出几种近似算法，发现$δ$从上方接近$2/3$时与其他边界情况行为不同。
AI 中文摘要
我们考虑一个名为$δ$-分散的连续设施选址问题。对于固定的$δ>0$，目标是在图上尽可能多地放置设施，使得两两之间的距离至少为$δ$，设施可位于图的顶点或边的内部。此问题可看作是著名独立集问题的连续版本。Grigoriev等人表明当$δ = 1/x$或$δ = 2/x$（$x$为自然数）时，$δ$-分散问题可在多项式时间内解决，否则为NP难问题。我们研究了$δ$-分散问题根据$δ$值的可近似性，对于$δ > 2$证明了多APX难，对于所有多项式时间不可解且$δ < 2$的情况证明了APX难。借助Hartmann等人的一个关于$δ$的平移定理，我们将注意力集中在区间$(2/3, 1)$和$(1, 2)$上，提供了几种近似算法，其近似因子在$δ$接近区间边界时趋近于1。令人惊讶的是，当$δ$从上方接近$2/3$时行为不同，在标准复杂性理论假设下，不可能构造一个当$δ$从上方接近$2/3$时近似因子趋近于1的近似算法。
英文摘要
We consider a continuous facility location problem called $δ$-Dispersion. For some fixed $δ> 0$, the goal is to place as many facilities on a graph as possible with pairwise distance at least $δ$. The facilities may be located on the vertices of the graph, or the interior of the edges. This problem can be interpreted as a continuous version of the well-known Independent Set problem. Its approximation behavior is very similar for large values of $δ$. Notably, Grigoriev et al. [Algorithmica 21] showed that $δ$-Dispersion is solvable in polynomial time when $δ= 1/x$ or $δ= 2/x$ for a natural number $x$ and NP-hard otherwise. We study the approximability of $δ$-Dispersion depending on the value of $δ$. For $δ> 2$, we show poly-APX-hardness, while for all $δ< 2$ that are not solvable in polynomial time we show APX-hardness. Thanks to a translation theorem for $δ$ due to Hartmann et al. [MFCS 22], we may focus our attention for approximation algorithms on the intervals $(2/3 , 1)$ and $(1, 2)$. We provide several approximation algorithms with an approximation factor approaching $1$ as $δ$ approaches one of the interval boundaries. Surprisingly, the behavior as $δ$ approaches $2/3$ from above is different: As our hardness reductions reveal, it is impossible (under standard complexity-theoretic assumptions) to construct an approximation algorithm with an approximation factor approaching $1$ as $δ$ approaches $2/3$ from above.
Comments16 pages, 3 figures, submitted to WAOA26
