On the complexity of Multipacking

TL;DR

We resolve the long-standing open question on the algorithmic complexity of Multipacking by proving NP-completeness for undirected graphs and $W[2]$-hardness when parameterized by the solution size $k$. The authors strengthen the hardness landscape by showing NP-completeness and $W[2]$-hardness for several graph classes, including chordal $\cap$ $\tfrac{1}{2}$-hyperbolic, bipartite, claw-free, regular, and CONV graphs, and provide an exact exponential-time algorithm with running time $O^*(1.58^n)$ that beats the naive $2^n$ barrier. The reductions rely on Hitting Set (and relatedly on Total Dominating Set) to establish intractability and ETH-based lower bounds, while the algorithmic result offers a practical approach for exact computation on general graphs. Together, these results delineate a clear boundary between tractable and intractable instances and spur further questions on parameterized and approximation approaches.

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 $O^*(1.58^n)$.

Figures (3)

• Figure 1: Inclusion diagram for graph classes mentioned in this paper (and related ones). If a class $A$ has a downward path to class $B$, then $B$ is a subclass of $A$. The Multipacking problem is NP-hard for the graph classes above the dashed (purple) straight-line and it is polynomial-time solvable for the graph classes below the dashed (purple) straight-line. Moreover, the Multipacking problem is W[2]-hard for the graph classes above the dashed (darkorange) curve. In this paper, we have discussed the hardness of the Multipacking problem for the graph classes in the colored (lightblue) block.
• Figure 3: Forbidden isometric subgraphs for chordal $\cap$$\frac{1}{2}$-hyperbolic graphs
• Figure 9: Rooted trees $H_1(k)$ and $H_2(k_1,k_2)$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 3
• Corollary 4
• Lemma 4
• Theorem 5: brinkmann2001hyperbolicity
• Lemma 5
• Lemma 5
• Lemma 5