Minimum Cost Source Location Problems with Flow Requirements

In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{\kappa}$\end{document} ), where the source location problems are to find a minimum-cost set S⊆V in a given graph G=(V,A) with a capacity function u:A→ℝ+ such that for each vertex v∈V, the connectivity from S to v (resp., from v to S) is at least a given demand d−(v) (resp., d+(v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54–68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of cln D for some constant c, unless every problem in NP has an O(Nlog log N)-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1+ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are Θ(ln ∑v∈V(d+(v)+d−(v)))-approximable.