连接半径和最小的圆集
Connecting a Set of Circles with Minimum Sum of Radii
摘要：
我们考虑的问题，分配半径给一组给定的点在平面上，这样所得到的一组圆是连接的，半径的总和是最小的。我们表明，如果一个连接树是多项式可解的问题。如果连通树是未知的，如果半径有上界，则问题是NP难的，否则是开放的。我们给近似保证各种多项式时间算法，描述上限和下限（这是匹配的情况下），提供多项式时间近似方案，并得出结论与实验结果和开放的问题。
英文摘要：
We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NP-hard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomial-time algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomial-time approximation schemes, and conclude with experimental results and open problems.
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。
