The Two-Squirrel Problem and Its Relatives

TL;DR

This paper introduces the Two-Squirrel problem, a variation of the star cover problem, proves its NP-hardness, explores related problems, and provides approximation algorithms with specific factors.

Contribution

It formulates the Two-Squirrel problem, proves its NP-hardness, analyzes related problems, and offers approximation algorithms with proven bounds.

Findings

Two-Squirrel problem is strongly NP-hard.

Approximation algorithms with factors 3.6402 and 4+ε for Two-MST and Two-TSP.

Polynomial-time solvable cases for Two-MST.

Abstract

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.