TY - GEN

T1 - Approximate k-Steiner forests via the Lagrangian relaxation technique with internal preprocessing

AU - Segev, Danny

AU - Segev, Gil

PY - 2006

Y1 - 2006

N2 - 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 Fscr; ⊆ 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 A-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} · log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands.

AB - 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 Fscr; ⊆ 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 A-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} · log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands.

UR - http://www.scopus.com/inward/record.url?scp=33750723122&partnerID=8YFLogxK

U2 - 10.1007/11841036_54

DO - 10.1007/11841036_54

M3 - Conference contribution

AN - SCOPUS:33750723122

SN - 3540388753

SN - 9783540388753

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 600

EP - 611

BT - Algorithms, ESA 2006 - 14th Annual European Symposium, Proceedings

PB - Springer Verlag

T2 - 14th Annual European Symposium on Algorithms, ESA 2006

Y2 - 11 September 2006 through 13 September 2006

ER -