# Approximating Some Network Design Problems with Node Costs

• Guy Kortsarz
• Zeev Nutov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5687)

## Abstract

We study several multi-criteria undirected network design problems with node costs and lengths with all problems related to the node costs 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 on a single path, so that ∑  v c v (f v ) is minimized, where f v the total amount of flow through v. In the Multicommodity Cost-Distance (mcd) problem we are also given lengths {ℓ(v):v ∈ V}, and seek a subgraph H of G that minimizes c(H) + ∑  s,t ∈ V d st ·ℓ H (s,t), where ℓ H (s,t) is the minimum ℓ-length of an st-path in H. The approximation for these two problems is equivalent up to a factor arbitrarily close to 2. We give an O(log3 n)-approximation algorithm for both problems for the case of demands polynomial in n. The previously best known approximation ratio for these problems was O(log4 n) [Chekuri et al., FOCS 2006] and [Chekuri et al., SODA 2007]. This technique seems quite robust and was already used in order to improve the ratio of Buy-at-bulk with protection (Antonakopoulos et al FOCS 2007) from log3 h to log2 h. See ?.

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, STOC 2001] computes a tree T with c(T) ≤ 2C and p(T) = Ω(opt/logn). We provide the first nontrivial lower bound and in fact provide a bicriteria lower bound on approximating this problem (which is stronger than the usual lower bound) by showing that the problem admits no better than Ω(1/(loglogn)) approximation assuming $$\mbox{NP}\not\subseteq \mbox{Quasi(P)}$$ even if the algorithm is allowed to violate the budget by any universal constant ρ. This disproves a conjecture of [Moss and Rabani, STOC 2001].

Another related to mbb problem 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 (T) ≤ L and c(T) minimum. We give an algorithm that computes a tree T with c(T) = O(log2 n) ·opt and diam (T) = O(logn) ·L. Previously, a polylogarithmic bicriteria approximation was known only for the case of edge costs and edge lengths.

## Keywords

Network design Node costs Multicommodity Buy at Bulk Covering tree Approximation algorithm Hardness of approximation

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