The Two-Squirrel Problem and Its Relatives

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_...

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Bibliographic Details
Main Authors Bereg, Sergey, Higashikawa, Yuya, Katoh, Naoki, Lafond, Manuel, Tokuni, Yuki, Zhu, Binhai
Format Journal Article
LanguageEnglish
Published 12.02.2023
Subjects
Computer Science - Computational Geometry
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Summary: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.
DOI:10.48550/arxiv.2302.05937