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 $2+o(1)$.