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 cv:v∈V of subadditive cost functions. For every s,t∈V we seek to send dst flow units from s to t, so that ∑vcv(fv) is minimized, where fv is the total amount of flow through v. It is shown in Andrews and Zhang (2002)  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∈Vdst·ℓH(s,t), where ℓH(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. 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)  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  with ρ=2. Our result disproves a conjecture of . 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.
Bibliographical noteFunding Information:
We thanks a referee for his/her insightful comments that significantly helped in improving the presentation of the paper. The first author was partially supported by NSF support grant award number 0829959.
- Approximation algorithm
- Multicommodity buy at bulk
- Network design
- Node costs