Approximating some network design problems with node costs

• Authors: Guy Kortsarz;Zeev Nutov
• Venue: Theoretical Computer Science
• Year: 2011

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"s"t:s,t@?V}, and a family {c"v:v@?V} of subadditive cost functions. For every s,t@?V we seek to send d"s"t 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-@e in the ratio, we may assume that each st-flow is unsplittable, namely, uses only one path. In the Multicommodity Cost-Distance (MCD) problem we are also given lengths {@?(v):v@?V}, and seek a subgraph H of G that minimizes c(H)+@?"s","t"@?"Vd"s"t@?@?"H(s,t), where @?"H(s,t) is the minimum @?-length of an st-path in H. The approximability of these two problems is equivalent up to a factor 2-@e[2]. We give an O(log^3n)-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(log^4n) (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)=@W(opt/logn). We provide the first nontrivial lower bound on approximation by proving that the problem admits no better than @W(1/(loglogn)) approximation assuming NP@?Quasi(P). This holds true even if the solution is allowed to violate the budget by a constant @r, as was done in [18] with @r=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 {@?(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"@?(T)@?L and c(T) minimum. We give an algorithm that computes a tree T with c(T)=O(log^2n)@?opt and diam"@?(T)=O(logn)@?L. Previously, a polylogarithmic bicriteria approximation was known only for the case of edge costs and edge lengths.