Multipacking on graphs and Euclidean metric space

TL;DR

This paper proves that the Multipacking problem is NP-complete and W[2]-hard for undirected graphs, explores its complexity in various graph classes, and studies geometric variants in Euclidean space, providing algorithms and hardness results.

Contribution

It resolves the open question of NP-completeness for undirected graphs and extends complexity results to multiple graph classes and geometric settings.

Findings

Multipacking is NP-complete for undirected graphs.

W[2]-hard when parameterized by solution size.

Polynomial-time algorithm for maximum 1-multipacking in .

Abstract

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 $\delta$-hyperbolic graphs. Moreover, we study the relationship between multipacking number and broadcast domination number for cactus, chordal, and $\delta$-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.