The Two-Squirrel Problem and Its Relatives

02/12/2023

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by   Sergey Bereg, et al.

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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+ε approximations respectively. Finally, we also show some interesting polynomial-time solvable cases for Two-MST.