Approximating the Shallow-Light Steiner Tree Problem When Cost and Delay are Linearly Dependent

Let G = (V, E) be a given graph, S ⊆ V be a terminal set, r ∈ S be the selected root. Assume that c : E → R⁺ and d : E → R⁺ are cost and delay functions on the edges respectively. The shallow-light Steiner tree (SLST) problem is to compute a minimum cost tree spanning all the terminals of S, such that the delay between r and every other terminal is bounded by a given delay constraint D ∈ R₀⁺. Since in real network, the cost and delay of a link are always related, this paper addresses two such special cases: the constrained Steiner tree (CST) problem, a special case of the SLST problem that c(e) = σd(e) for every edge, and the constrained spanning tree (CPT) problem, a further special case of the CST problem when S = V . This paper first shows that even when c(e) = d(e), the CPT problem is NP-hard, and admits no (1 + ∈,γln(|V|)-approximation algorithm for some fixed γ > 0 and any ∈ <; |V |+|E|+1. The inapproximability result can be applied to the1 CST problem immediately. Based on the above observation of the hardness to develop a single factor approximation algorithm, we give an approximation algorithm with a bifactor ratio of (ρ, 1.39 + 2.78/ρ-1) for the CST problem, where 1.39 is the best approximation ratio for the minimum Steiner tree problem in the current state of the art. As a consequence, for the applications where cost and delay are of equal importance, an approximation with bifactor (2.87, 2.87)for CST can be immediately obtained by setting ρ = 1.39+ 2.78/ρ-1.This indicates that the SLST problem admits approximation algorithms with constant bifactor ratio, when the cost and delay are linearly dependent.