Approximating Some Network Design Problems with Node Costs

• Authors: Guy Kortsarz;Zeev Nutov
• Venue: APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
• Year: 2009

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 {***(v ):v *** V }, and seek a subgraph H of G that minimizes c (H ) + *** s ,t *** V d st ·*** H (s ,t ), where *** H (s ,t ) is the minimum ***-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 . 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 ***(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 {***(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 and c (T ) minimum. We give an algorithm that computes a tree T with and . Previously, a polylogarithmic bicriteria approximation was known only for the case of edge costs and edge lengths.