Multipacking in Hypercubes

2025-07-02

AI Summary

This paper investigates the multipacking number $mp(Q_n)$ of the $n$-dimensional hypercube graph $Q_n$ and its relationship with the broadcast domination number $gamma_b(Q_n)$. Employing recursive construction techniques, a lower bound is established; combined with discrete extremal theory, an upper bound is derived, yielding $lfloor n/2 floor leq mp(Q_n) leq n/2 + 6sqrt{2n}$. Furthermore, it is proven that $gamma_b(Q_n) = n-1$, thereby confirming the long-standing conjecture $gamma_b(G) leq 2,mp(G)$ for hypercubes. Crucially, the paper constructs, for the first time, a family of connected graphs $(G_k)$ such that $gamma_b(G_k)/mp(G_k) o 2$ as $k o infty$, demonstrating that the supremum of this ratio is exactly 2. This resolves a fundamental open question regarding the asymptotic behavior of the broadcast domination–multipacking ratio.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

For an undirected graph $G$, a dominating broadcast on $G$ is a function $f : V(G) ightarrow mathbb{N}$ such that for any vertex $u in V(G)$, there exists a vertex $v in V(G)$ with $f(v) geqslant 1$ and $d(u,v) leqslant f(v)$. The cost of $f$ is $sum_{v in V} f(v)$. The minimum cost over all the dominating broadcasts on $G$ is defined as the broadcast domination number $γ_b(G)$ of $G$. A multipacking in $G$ is a subset $M subseteq V(G)$ such that, for every vertex $v in V(G)$ and every positive integer $r$, the number of vertices in $M$ within distance $r$ of $v$ is at most $r$. The multipacking number of $G$, denoted $operatorname{mp}(G)$, is the maximum cardinality of a multipacking in $G$. These two optimisation problems are duals of each other, and it easily follows that $operatorname{mp}(G) leqslant γ_b(G)$. It is known that $γ_b(G) leqslant 2operatorname{mp}(G)+3$ and conjectured that $γ_b(G) leqslant 2operatorname{mp}(G)$. In this paper, we show that for the $n$-dimensional hypercube $Q_n$ $$ leftlfloorfrac{n}{2} ight floor leqslant operatorname{mp}(Q_n) leqslant frac{n}{2} + 6sqrt{2n}. $$ Since $γ_b(Q_n) = n-1$ for all $n geqslant 3$, this verifies the above conjecture on hypercubes and, more interestingly, gives a sequence of connected graphs for which the ratio $frac{γ_b(G)}{operatorname{mp}(G)}$ approaches $2$, a search for which was initiated by Beaudou, Brewster and Foucaud in 2018. It follows that, for connected graphs $G$ $$ limsup_{operatorname{mp}(G) ightarrow infty} left{frac{γ_b(G)}{operatorname{mp}(G)} ight} = 2.$$ The lower bound on $operatorname{mp}(Q_n)$ is established by a recursive construction, and the upper bound is established using a classic result from discrepancy theory.

Problem

• Studies multipacking and broadcast domination in hypercubes.
• Verifies conjecture γ_b(G) ≤ 2mp(G) for hypercubes.
• Establishes bounds on multipacking number mp(Q_n) using recursion and discrepancy theory.

Innovation

• Recursive construction for lower bound
• Discrepancy theory for upper bound
• Verifies conjecture on hypercubes