循环图与子集的GCD和LCM
Circulant graphs and GCD and LCM of Subsets
摘要：
给定两个整数集合$A$和$B$，我们考虑寻找一个集合$S \subseteq A$的最小可能基数，使得$S \cup B$的元素的最大公约数等于$A \cup B$的元素的最大公约数. $B = \emptyset$和$\#B = 1$的特殊情况是特别有趣的，并且与图论有一些联系。我们还考虑了元素的最小公倍数的相应问题。我们通过将这些问题与最小覆盖问题联系起来，建立了这些问题的NP完全性和近似结果。
英文摘要：
Given two sets $A$ and $B$ of integers, we consider the problem of finding a set $S \subseteq A$ of the smallest possible cardinality such the greatest common divisor of the elements of $S \cup B$ equals that of those of $A \cup B$. The particular cases of $B = \emptyset$ and $\#B = 1$ are of special interest and have some links with graph theory. We also consider the corresponding question for the least common multiple of the elements. We establish NP-completeness and approximation results for these problems by relating them to the Minimum Cover Problem.
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。
