Approximation algorithms for quickest spanning tree problems

Let G = (V, E) be an undirected multigraph with a special vertex root ∈ V, and where each edge e ∈ E is endowed with a length l(e) ≥ 0 and a capacity c(e) > 0. For a path P that connects u and v, the transmission time of P is defined as t(P) = ∑_e∈Pl(e) + max _e∈P(1/c(e)). For a spanning tree T, let P_{u, v} ^T be the unique u-v path in T. The QUICKEST RADIUS SPANNING TREE PROBLEM is to find a spanning tree T of G such that max_v∈V t(P_{root, v}^T) is minimized. In this paper we present a 2-approximation algorithm for this problem, and show that unless P = N P, there is no approximation algorithm with a performance guarantee of 2 - ε for any ε > 0. The QUICKEST DIAMETER SPANNING TREE PROBLEM is to find a spanning tree T of G such that max_{u, v∈V} t(P_{u, v} ^T) is minimized. We present a 3/2-approximation to this problem, and prove that unless P = N P there is no approximation algorithm with a performance guarantee of 3/2 - ε for any ε > 0.