Capacitated Network Design Problems: Hardness, Approximation Algorithms, and Connections to Group Steiner Tree

We design combinatorial approximation algorithms for the Capacitated Steiner Network (Cap-SN) problem and the Capacitated Multicommodity Flow (Cap-MCF) problem. These two problems entail satisfying connectivity requirements when edges have costs and hard capacities. In Cap-SN, the flow has to be supported separately for each commodity while in Cap-MCF, the flow has to be sent simultaneously for all commodities. We show that the Group Steiner problem on trees ([12]) is a special case of both problems. This implies the first polylogarithmic lower bound for these problems by [17]. We then give various approximations to special cases of the problems. We generalize the well known Source location problem (see for example [19]), to a natural problem called the Connected Rent or Buy Source Location problem. We show that this problem is a a simplification of Cap-SN and CapMCF and a generalization of Group Steiner on general graphs. We use Group Steiner Tree techniques, and more sophisticated techniques, to derive log n approximation for the Connected Rent or Buy Source Location problem which is close to the best approximation known for Group Steiner on general graphs. Another special case we study is as follows. Given a bipartite graph G = (A∪B,E) and an integer k > 0, find A′ ⊆ A and B′ ⊆ B of minimum total size |A′|+ |B′| such that there exist k edge-disjoint paths in G from vertices in A′ to vertices in B′. This problem is a special case of the Steiner Network problem with vertex costs [20]. In [20] Nutov asked the open question if the Steiner network problem with vertex costs admits an o(k) ratio. We give an o(k) approximation for this special case, which could be a step toward resolving the open question of Nutov. We provide an O( √ k log k) approximation ratio for the problem. We also show that we can compute a solution of optimum value, while being able to route O(k/polylog n) flow, where n is ? Part of this work was done at DIMACS. We thank DIMACS for their hospitality. ?? Supported in part by NSF CAREER award 1053605, ONR YIP award N000141110662, DARPA/AFRL award FA8650-11-1-7162, and University of Maryland Research and Scholarship Award (RASA). The author is also with AT&T Labs– Research, Florham Park, NJ. ? ? ? Supported in part by NSF grant number 434923. the number of vertices in G. The final special case of Cap-SN and CapMCF that we study is called the Unbalanced-P2P problem. Besides its practical applications to shift design problems [8], it generalizes many problems such as k-Steiner tree, Steiner Forest, and Point-to-Point Connection. We give a combinatorial logarithmic approximation algorithm for this problem.