On the complexity of Multipacking

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 $M$, that is, there are at most $r$ vertices in $M$ at a distance at most $r$ from $v$ in $G$. The \textsc{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 \textsc{Multipacking} problem is \textsc{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 [\textit{Algorithmica} 2021] made a step towards solving the open question by showing that the \textsc{Multipacking} problem is \textsc{NP}-complete for directed graphs and it is \textsc{W[1]}-hard when parameterized by the solution size. We prove that the \textsc{Multipacking} problem is \textsc{NP}-complete and \textsc{W[2]}-hard on undirected graphs when parameterized by the solution size. These hardness results also hold for chordal, bipartite, claw-free, and chordal $\cap$ $\frac{1}{2}$-hyperbolic graphs. Chordal $\cap$ $\frac{1}{2}$-hyperbolic graph class contains strongly chordal graphs, where the problem is polynomial-time solvable. We further establish \textsc{NP}-completeness on regular and CONV graphs (intersection graphs of convex sets in the plane). On the positive side, we present an exact $O(1.53^n)$-time algorithm for general $n$-vertex graphs, breaking the $2^n$ barrier.