Broadcast Domination Number is at Most Twice the Multipacking Number

Abstract: For a graph $G = (V, E)$ with a vertex set $V$ and an edge set $E$, a function $f : V \rightarrow {0, 1, 2, . . . , diam(G)}$ is called a \emph{broadcast} on $G$. For each vertex $u \in V$, if there exists a vertex $v$ in $G$ (possibly, $u = v$) such that $ f (v) > 0 $ and $d(u, v) \leq f (v)$, then $f$ is called a dominating broadcast on $G$. The cost of the dominating broadcast $f$ is the quantity $\sum_{v\in V}f(v)$. The minimum cost of a dominating broadcast is the broadcast domination number of $G$, denoted by $γ_{b}(G)$. A multipacking is a set $M \subseteq V$ in a graph $G = (V, E)$ 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 number of $G$ is the maximum cardinality of a multipacking of $G$ and is denoted by $mp(G)$. It is known that $mp(G)\leqγ_b(G)$. In 2014, Hartnell and Mynhardt proved that $γ_b(G)\leq 3mp(G)-2$ whenever $mp(G)\geq2$. In 2019, Beaudou, Brewster, and Foucaud improved this bound to $γ_b(G)\leq 2 mp(G)+3$ and conjectured that $γ_b(G)\leq 2 mp(G)$. We solve their conjecture by proving that $γ_b(G)\leq 2 mp(G)$ for every graph $G$. Our proof is constructive and yields a polynomial-time $2$-approximation algorithm for Maximum Multipacking problem which improves the earlier approximation factor $2+o(1)$.