In this paper, we start with a variation of the star cover problem called the Two-Squirrel problem. Given a set $P$ of $2n$ points in the plane, and two sites $c_1$ and $c_2$, compute two $n$-stars $S_1$ and $S_2$ centered at $c_1$ and $c_2$ respectively such that the maximum weight of $S_1$ and $S_2$ is minimized. This problem is strongly NP-hard by a reduction from Equal-size Set-Partition with Rationals. Then we consider two variations of the Two-Squirrel problem, namely the Two-MST and Two-TSP problem, which are both NP-hard. The NP-hardness for the latter is obvious while the former needs a non-trivial reduction from Equal-size Set-Partition with Rationals. In terms of approximation algorithms, for Two-MST and Two-TSP we give factor 3.6402 and $4+\varepsilon$ approximations respectively. Finally, we also show some interesting polynomial-time solvable cases for Two-MST.

翻译：本文首先研究星形覆盖问题的一个变体——双松鼠问题。给定平面上2n个点构成的集合P，以及两个站点c₁和c₂，要求分别以c₁和c₂为中心构造两个n星形S₁和S₂，使得S₁和S₂的最大权重最小化。通过从有理数等势集分割问题进行归约，证明该问题为强NP难问题。随后考虑双松鼠问题的两个变体——双最小生成树问题与双旅行商问题，二者均为NP难问题。后者NP难性显然，而前者需通过从有理数等势集分割问题进行非平凡归约。在近似算法方面，针对双最小生成树问题与双旅行商问题，分别给出因子为3.6402和4+ε的近似比。最后，本文还展示了双最小生成树问题若干有趣的多项式时间可解情形。