For a graph G=(V,E) with vertex set V and edge set E, a
function f:Vβ{0,1,2,...,diam(G)} is called a
\emph{broadcast} on G. For each vertex uβV, if there exists a
vertex v in G (possibly, u=v) such that f(v)>0 and d(u,v)β€f(v), then f is called a \textit{dominating broadcast} on G.
The \textit{cost} of the dominating broadcast f is the quantity βvβVβf(v). The minimum cost of a dominating broadcast is the \textit{broadcast
domination number} of G, denoted by Ξ³bβ(G). A
\textit{multipacking} is a set SβV in a graph G=(V,E) such
that for every vertex vβV and for every integer rβ₯1, the ball
of radius r around v contains at most r vertices of S, that is,
there are at most r vertices in S at a distance at most r from v in G. The \textit{multipacking number} of G is the maximum
cardinality of a multipacking of G and is denoted by mp(G). We show
that, for any cactus graph G, Ξ³bβ(G)β€23βmp(G)+211β. We also show that Ξ³bβ(G)βmp(G) can be
arbitrarily large for cactus graphs by constructing an infinite family of
cactus graphs such that the ratio Ξ³bβ(G)/mp(G)=4/3, with mp(G)
arbitrarily large. This result shows that, for cactus graphs, we cannot improve
the bound Ξ³bβ(G)β€23βmp(G)+211β to a bound in the
form Ξ³bβ(G)β€c1ββ mp(G)+c2β, for any constant c1β<4/3 and
c2β. Moreover, we provide an O(n)-time algorithm to construct a
multipacking of G of size at least 32βmp(G)β311β, where
n is the number of vertices of the graph G