Approximating some network design problems with node costs

We study several multi-criteria undirected network design problems with node costs and lengths. All these problems are related to the Multicommodity Buy at Bulk (MBB) problem in which we are given a graph G=(V,E), demands d _st:s, t ∈ V, and a family c_v:v∈V of subadditive cost functions. For every s,t∈V we seek to send d_st flow units from s to t, so that ∑_vc_v(f_v) is minimized, where f_v is the total amount of flow through v. It is shown in Andrews and Zhang (2002) [2] that with a loss of 2-ε in the ratio, we may assume that each st-flow is unsplittable, namely, uses only one path. In the Multicommodity CostDistance (MCD) problem we are also given lengths ℓ(v):v∈V, and seek a subgraph H of G that minimizes c(H)+∑_s,t∈Vd_st·ℓ_H(s,t), where ℓ_H(s,t) is the minimum ℓ-length of an st-path in H. The approximability of these two problems is equivalent up to a factor 2-ε[2]. We give an O(log³n)-approximation algorithm for both problems for the case of the demands polynomial in n. The previously best known approximation ratio for these problems was O(log⁴n) (Chekuri et al., 2006, 2007) [5,6]. We also consider the Maximum Covering Tree (MaxCT) problem which is closely related to MBB: given a graph G=(V,E), costs c(v):v∈V, profits p(v):v∈V, and a bound C, find a subtree T of G with c(T)≤C and p(T) maximum. The best known approximation algorithm for MaxCT (Moss and Rabani, 2001) [18] computes a tree T with c(T)≤2C and p(T)=Ω(optlogn). We provide the first nontrivial lower bound on approximation by proving that the problem admits no better than Ω(1(loglogn)) approximation assuming NP⊈Quasi(P). This holds true even if the solution is allowed to violate the budget by a constant ρ, as was done in [18] with ρ=2. Our result disproves a conjecture of [18]. Another problem related to MBB is the Shallow Light Steiner Tree (SLST) problem, in which we are given a graph G=(V,E), costs c(v):v∈V, lengths ℓ(v):v∈V, a set U⊆V of terminals, and a bound L. The goal is to find a subtree T of G containing U with diam ℓ (T)≤L and c(T) minimum. We give an algorithm that computes a tree T with c(T)=O(log²n)·opt and diam ℓ (T)=O(logn)·L. Previously, a polylogarithmic bicriteria approximation was known only for the case of edge costs and edge lengths.