Approximation Algorithms for Quickest Spanning Tree Problems

Abstract

Let G=(V,E) be an undirected multi-graph 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 $t(P)=\sum_{e \in P} l(e) + \max_{e \in P} {1 \over c(e)}$. 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 $\max _{v \in V} t(P^T_{root,v})$ is minimized. In this paper we present a 2-approximation algorithm for this problem, and show that unless P =NP, there is no approximation algorithm with 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 $\max_{u,v \in V} t(P^T_{u,v})$ is minimized. We present a ${3 \over 2}$-approximation to this problem, and prove that unless P=NP there is no approximation algorithm with performance guarantee of ${3 \over 2}-\epsilon$ for any ε >0.