Distance Approximating Trees: Complexity and Algorithms

Let Δ≥ 1 and δ≥ 0 be real numbers. A tree T=(V,E′) is a distance (Δ,δ)–approximating tree of a graph G=(V,E) if dH(u,v)≤Δ dG(u,v)+δ and dG(u,v)≤Δ dH(u,v)+δ hold for every u,v∈ V. The distance (Δ,δ)-approximating tree problem asks for a given graph G to decide whether G has a distance (Δ,δ)-approximating tree. In this paper, we consider unweighted graphs and show that the distance (Δ,0)-approximating tree problem is NP-complete for any Δ≥ 5 and the distance (1,1)-approximating tree problem is polynomial time solvable.

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