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arXiv 2007.11476cs.DScs.CG
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基于LP舍入的上下界近似覆盖

Approximate Covering with Lower and Upper Bounds via LP Rounding

Sayan Bandyapadhyay, Aniket Basu Roy

备注：
There is an error in the algorithm for LUC in Section 3. The proof of Lemma 5 does not hold
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摘要：

本文研究了上下界覆盖（LUC）问题，给出了$n$点的集合$P$，球的集合$\mathcal{B}$，参数$L$和$U$ .目标是找到最小尺寸的子集$\mathcal{B}'\subseteq \mathcal{B}$和$P$中的点到$\mathcal{B}'$的分配，使得每个点$p\in P$被分配给包含$p$的球，并且对于每个球$B_i\in \mathcal{B}'$，至少$L$和至多$U$点被分配给$B_i$。我们得到了一个LP舍入为基础的常数近似LUC违反下限和上限的限制，小常数因子和扩大的球再次一个小常数因子。类似的结果是已知的覆盖问题，只有上限约束。我们还证明了，只有下界约束，上述结果可以得到没有任何下界破坏。覆盖问题与设施选址问题有着密切的联系。我们注意到，已知的常数近似相应的下限和上限设施选址问题，违反了一个常数因子的下限和上限约束。

英文摘要：

In this paper, we study the lower- and upper-bounded covering (LUC) problem, where we are given a set $P$ of $n$ points, a collection $\mathcal{B}$ of balls, and parameters $L$ and $U$. The goal is to find a minimum-sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an assignment of the points in $P$ to $\mathcal{B}'$, such that each point $p\in P$ is assigned to a ball that contains $p$ and for each ball $B_i\in \mathcal{B}'$, at least $L$ and at most $U$ points are assigned to $B_i$. We obtain an LP rounding based constant approximation for LUC by violating the lower and upper bound constraints by small constant factors and expanding the balls by again a small constant factor. Similar results were known before for covering problems with only the upper bound constraint. We also show that with only the lower bound constraint, the above result can be obtained without any lower bound violation. Covering problems have close connections with facility location problems. We note that the known constant-approximation for the corresponding lower- and upper-bounded facility location problem, violates the lower and upper bound constraints by a constant factor.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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