Broadcast domination and multipacking: bounds and the integrality gap
Résumé
The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence, respectively. In 2001, Erwin introduced \emph{broadcast domination} in graphs, a covering problem using balls of various radii, where the cost of a ball is its radius. The minimum cost of a dominating broadcast in a graph $G$ is denoted by $\gamma_b(G)$. The dual (in the sense of linear programming) of broadcast domination is \emph{multipacking}: a multipacking is a set $P \subseteq V(G)$ such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $P$. The maximum size of a multipacking in a graph $G$ is denoted by $mp(G)$. Naturally, $mp(G) \leq \gamma_b(G)$. Hartnell and Mynhardt proved that $\gamma_b(G) \leq 3 mp(G) - 2$ (whenever $mp(G)\geq 2$). In this paper, we show that $\gamma_b(G) \leq 2mp(G) + 3$. Moreover, we conjecture that this can be improved to $\gamma_b(G) \leq 2mp(G)$ (which would be sharp).