Approximation algorithm for finding multipacking on Cactus
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For a graph G = (V, E) with vertex set V and edge set E, a function f : V →{0, 1, 2, . . . , diam(G)} is called a broadcast on G. For each vertex u ∈ V, if there exists a vertex v in G (possibly, u = v) such that f (v) > 0 and d(u, v) ≤ f (v), then f is called a dominating broadcast on G. The cost of the dominating broadcast f is the quantity ∑_v∈ Vf(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 S ⊆ V in a graph G = (V, E) such that for every vertex v ∈ V and for every integer r ≥ 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 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, γ_b(G)≤3/2mp(G)+11/2. We also show that γ_b(G)-mp(G) can be arbitrarily large for cactus graphs by constructing an infinite family of cactus graphs such that the ratio γ_b(G)/mp(G)=4/3, with mp(G) arbitrarily large. This result shows that, for cactus graphs, we cannot improve the bound γ_b(G)≤3/2mp(G)+11/2 to a bound in the form γ_b(G)≤ c_1· mp(G)+c_2, for any constant c_1<4/3 and c_2. Moreover, we provide an O(n)-time algorithm to construct a multipacking of G of size at least 2/3mp(G)-11/3, where n is the number of vertices of the graph G.
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