We study the task of gathering k energy-constrained mobile agents in an undirected edge-weighted graph. Each agent is initially placed on an arbitrary node and has a limited amount of energy, which constrains the distance it can move. Since this may render gathering at a single point impossible, we study three variants of near-gathering:

The goal is to move the agents into a configuration that minimizes either (i) the radius of a ball containing all agents, (ii) the maximum distance between any two agents, or (iii) the average distance between the agents. We prove that (i) is polynomial-time solvable, (ii) has a polynomial-time 2-approximation with a matching NP-hardness lower bound, while (iii) admits a polynomial-time $2(1-\frac{1}{k})$-approximation, but no FPTAS, unless $\text{P}=\text{NP}$. We extend some of our results to additive approximation.

我们研究了在无向边加权图中收集k个能量受限的移动代理的任务。每个代理最初都放置在任意节点上，并且能量有限，这限制了它可以移动的距离。由于这可能使在单个点上的聚集变得不可能，因此我们研究了近距离聚集的三种变体：

目标是将代理移动到最小化（i）包含所有代理的球的半径，（ii）任何两个代理之间的最大距离或（iii）代理之间的平均距离的配置。我们证明（i）是多项式时间可解的（ii）具有多项式时间2逼近且具有匹配的NP硬度下限，而（iii）允许多项式时间$2（\mathrm{1个}-\frac{\mathrm{1个}}{ķ}）$-近似值，但没有FPTAS，除非 $\text{P}=\text{NP}$。我们将一些结果扩展到加法近似。