Approximation to the Minimum Cost Edge Installation Problem

Abstract

We consider the minimum cost edge installation problem (MCEI) in a graph G = (V,E) with edge weight w(e) ≥ 0, e ∈ E. We are given a vertex s ∈ V designated as a sink, an edge capacity λ> 0, and a source set S ⊆ V with demand q(v) ∈ [0,λ], v ∈ S. For any edge e ∈ E, we are allowed to install an integer number h(e) of copies of e. The MCEI asks to send demand q(v) from each source v ∈ S along a single path P _v to the sink s. A set of such paths can pass through a single copy of an edge in G as long as the total demand along the paths does not exceed the edge capacity λ. The objective is to find a set ${\mathcal P}=\{P_v\mid v\in S\}$ of paths of G that minimizes the installing cost ∑ _e ∈ E h(e) w(e). In this paper, we propose a $(15/8+\rho_{\mbox{\tiny{\sc ST}}})$-approximation algorithm to the MCEI, where $\rho_{\mbox{\tiny{\sc ST}}}$ is any approximation ratio achievable for the Steiner tree problem.