Approximate k-Steiner forests via the Lagrangian relaxation technique with internal preprocessing

• Authors: Danny Segev;Gil Segev
• Venue: ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
• Year: 2006

An instance of the k-Steiner forest problem consists of an undirected graph G = (V,E), the edges of which are associated with non-negative costs, and a collection D ={(si,ti): 1 ≤ i ≤ d} of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest F ⊆ G connects a demand (si, ti) when it contains an si-ti path. Given a requirement parameter K ≤ |D|, the goal is to find a minimum cost forest that connects at least k demands in D. This problem has recently been studied by Hajiaghayi and Jain [SODA'06], whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k-subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research. In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of O(min{n2/3, √d} ċ log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands.