We study the problem of covering the maximum number of vertices in a graph by a collection of vertex-disjoint stars, each with a number of satellites in a given interval $[k, \ell]$, where $1 \le k < \ell$ and $\ell$ can be infinity. This is referred to as sequential {\sc $[k, \ell]$-Star Packing} problem. It is solvable in polynomial time when $k = 1$, but becomes strongly NP-hard when $k \ge 2$. In this paper, we propose either the first or an improved approximation algorithm for the following four sequential settings: 1) a $\frac {k+1}2$-approximation algorithm when $k \ge 3$ and $\ell = \infty$, improving the previous best ratio of $\frac {(k+1)^2}{2k+1}$; 2) a $\frac 43$-approximation algorithm when $k = 2$ and $\ell = \infty$, improving the previous best ratio of $\frac 32$; 3) the first $(1 + \frac \ell{\ell+1})$-approximation algorithm when $2 = k < \ell$; and 4) the first $(1 + \max\left\{\frac {k-1}2, \frac {(k+1) \ell}{3 (\ell+1)}\right\})$-approximation algorithm when $3 \le k < \ell$. Besides the main algorithmic techniques being local search coupled with amortized analysis, we observe augmenting configurations to bridge two distant neighborhoods for a local improvement operation. Additionally, the problem has been shown APX-hard when $k \ge 3$; we prove its APX-hardness for the last remaining case where $k = 2$.

翻译：我们研究用顶点不相交的星形图集合覆盖图中最大数量顶点的问题，其中每个星形图的卫星数量位于给定区间$[k, \ell]$内，且$1 \le k < \ell$，$\ell$可以为无穷大。这被称为顺序{\sc $[k, \ell]$-星形图打包}问题。当$k = 1$时，该问题可在多项式时间内求解，但当$k \ge 2$时，变为强NP难问题。本文针对以下四种顺序设置，提出了首个或改进的近似算法：1) 当$k \ge 3$且$\ell = \infty$时，一个$\frac {k+1}2$-近似算法，改进了此前$\frac {(k+1)^2}{2k+1}$的最佳比率；2) 当$k = 2$且$\ell = \infty$时，一个$\frac 43$-近似算法，改进了此前$\frac 32$的最佳比率；3) 当$2 = k < \ell$时，首个$(1 + \frac \ell{\ell+1})$-近似算法；4) 当$3 \le k < \ell$时，首个$(1 + \max\left\{\frac {k-1}2, \frac {(k+1) \ell}{3 (\ell+1)}\right\})$-近似算法。除主要算法技术采用局部搜索结合摊销分析外，我们观察到增广配置可桥接两个远距离邻域以实现局部改进操作。此外，该问题在$k \ge 3$时已被证明为APX难；我们证明其在最后一个剩余情况$k = 2$下也为APX难。