On the complexity of Multipacking

2026-02-08

AI Summary

This study investigates the computational complexity of the Multipacking problem in undirected graphs: given a graph \( G \) and an integer \( k \), determine whether there exists a vertex subset \( S \) of size at least \( k \) such that, for every vertex \( v \) and radius \( r \), the ball of radius \( r \) centered at \( v \) contains at most \( r \) vertices from \( S \). Through a polynomial-time reduction, the authors establish for the first time that the problem is NP-complete on general undirected graphs and W[2]-hard when parameterized by solution size. These hardness results are further extended to several important graph classes, including chordal graphs, bipartite graphs, and claw-free graphs. Additionally, the paper presents an exact exponential algorithm running in \( O^*(1.58^n) \) time, improving upon the trivial \( O^*(2^n) \) enumeration bound.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

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 $O^*(1.58^n)$.

Problem

• Multipacking
• NP-completeness
• parameterized complexity
• graph classes
• computational complexity

Innovation

• Multipacking
• NP-completeness
• parameterized complexity
• exact exponential algorithm
• graph classes