A 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 $M$, that is, there are at most $r$ vertices in $M$ at a distance at most $r$ from $v$ in $G$. The Multipacking problem asks whether a graph contains a multipacking of size at least $k$. For more than a decade, it remained an open question whether the Multipacking problem is NP-complete or solvable in polynomial time. Whereas the problem is known to be polynomial-time solvable for certain graph classes (e.g., strongly chordal graphs, grids, etc). Foucaud, Gras, Perez, and Sikora [Algorithmica 2021] made a step towards solving the open question by showing that the Multipacking problem is NP-complete for directed graphs and it is W[1]-hard when parameterized by the solution size. In this paper, we prove that the Multipacking problem is NP-complete for undirected graphs, which answers the open question. Moreover, the problem is W[2]-hard for undirected graphs when parameterized by the solution size. Furthermore, we have shown that the problem is NP-complete and W[2]-hard (when parameterized by the solution size) even for various subclasses: chordal, bipartite, and claw-free graphs. Whereas, it is NP-complete for regular, and CONV graphs (intersection graphs of convex sets in the plane). Additionally, the problem is NP-complete and W[2]-hard (when parameterized by the solution size) for chordal $\cap$ $\frac{1}{2}$-hyperbolic graphs, which is a superclass of strongly chordal graphs where the problem is polynomial-time solvable. On the positive side, we present an exact exponential-time algorithm for the Multipacking problem on $n$-vertex general graphs, which breaks the $2^n$ barrier by achieving a running time of $O^*(1.58^n)$.

翻译：在无向图$G=(V,E)$中，多重装填是指顶点子集$M\subseteq V$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$r$的球中至多包含$M$的$r$个顶点，即$G$中与$v$距离不超过$r$的$M$中顶点数不超过$r$。多重装填问题要求判定给定图是否包含大小至少为$k$的多重装填集。十余年来，该问题属于NP完全问题还是多项式时间可解问题一直悬而未决。尽管已知该问题在某些图类（如强弦图、网格图等）上具有多项式时间算法。Foucaud、Gras、Perez和Sikora [Algorithmica 2021] 通过证明有向图上的多重装填问题是NP完全的，且以解大小为参数时是W[1]-难的，向解决这一开放问题迈出了重要一步。本文证明了无向图上的多重装填问题是NP完全的，从而彻底解答了这一开放问题。此外，对于无向图，以解大小为参数时该问题甚至是W[2]-难的。进一步地，我们证明了该问题在多个子类上——包括弦图、二分图和无爪图——同样是NP完全且以解大小为参数时是W[2]-难的。而对于正则图和CONV图（平面上凸集交图），该问题也是NP完全的。特别值得注意的是，对于弦图与$\frac{1}{2}$-双曲图的交图（这是强弦图的超类，而强弦图上该问题是多项式时间可解的），该问题同样是NP完全且以解大小为参数时是W[2]-难的。在积极方面，我们提出了针对$n$顶点一般图的多重装填问题的精确指数时间算法，该算法以$O^*(1.58^n)$的运行时间突破了$2^n$的屏障。