A \emph{multipacking} in an undirected graph $G=(V,E)$ is a set $M\subseteq V$ such that for every vertex $v\in V$ and for every integer $r\geq 1$, the ball of radius $ r $ around $ v $ contains at most $r$ vertices of $M$. The \textsc{Multipacking} problem asks whether a graph contains a multipacking of size at least $k$. For more than a decade, it remained open whether \textsc{Multipacking} is \textsc{NP-complete} or polynomial-time solvable, although it is known to be polynomial-time solvable for some classes (e.g., strongly chordal graphs and grids). Foucaud, Gras, Perez, and Sikora [\textit{Algorithmica} 2021] showed it is \textsc{NP-complete} for directed graphs and \textsc{W[1]-hard} when parameterized by the solution size. We resolve the open question by proving \textsc{Multipacking} is \textsc{NP-complete} for undirected graphs and \textsc{W[2]-hard} when parameterized by the solution size. Furthermore, we show it remains \textsc{NP-complete} and \textsc{W[2]-hard} even for chordal, bipartite, claw-free, regular, CONV, and chordal$\cap\frac{1}{2}$-hyperbolic graphs (a superclass of strongly chordal graphs), and we provide approximation algorithms for cactus, chordal, and $δ$-hyperbolic graphs. Moreover, we study the relationship between multipacking number and broadcast domination number for cactus, chordal, and $δ$-hyperbolic graphs. Further, we prove that for all $r\geq 2$, \textsc{$r$-Multipacking} is \textsc{NP-complete} even for planar bipartite graphs with bounded degree, and also for bounded-diameter chordal and bounded-diameter bipartite graphs. For geometric variants, in $\mathbb{R}^2$ a maximum $1$-multipacking can be computed in polynomial time, but computing a maximum $2$-multipacking is \textsc{NP-hard}, and we provide approximation and parameterized algorithms for the $2$-multipacking problem.

翻译：在无向图$G=(V,E)$中，\emph{多重装填}是指一个顶点子集$M\subseteq V$，满足对于任意顶点$v\in V$和任意整数$r\geq 1$，以$v$为中心、半径为$r$的球内至多包含$M$中的$r$个顶点。\textsc{多重装填}问题旨在判定一个图是否包含大小至少为$k$的多重装填集。十余年来，该问题究竟是\textsc{NP完全}还是多项式时间可解一直悬而未决，尽管已知其在某些图类（如强弦图和网格图）上存在多项式时间算法。Foucaud、Gras、Perez和Sikora在[\textit{Algorithmica} 2021]中证明了该问题在有向图上是\textsc{NP完全的}，且在以解大小为参数时是\textsc{W[1]-难的}。本文通过证明无向图上的\textsc{多重装填}问题是\textsc{NP完全的}，且在以解大小为参数时是\textsc{W[2]-难的}，从而解决了这一开放性问题。此外，我们证明该问题即使在弦图、二分图、无爪图、正则图、CONV图以及弦图$\cap\frac{1}{2}$-双曲图（强弦图的超类）上仍然保持\textsc{NP完全性}与\textsc{W[2]-难性}，并针对仙人掌图、弦图和$δ$-双曲图给出了近似算法。进一步，我们研究了仙人掌图、弦图和$δ$-双曲图的多重装填数与广播控制数之间的关系。此外，我们证明对于所有$r\geq 2$，\textsc{$r$-多重装填}问题即使在有界度平面二分图、有界直径弦图和有界直径二分图上也是\textsc{NP完全的}。对于几何变体问题，在$\mathbb{R}^2$中最大$1$-多重装填可在多项式时间内计算，但计算最大$2$-多重装填是\textsc{NP难的}，我们针对$2$-多重装填问题提出了近似算法与参数化算法。