Node-weighted Steiner tree and group Steiner tree in planar graphs
Author(s)Demaine, Erik D.; Hajiaghayi, Mohammad Taghi; Klein, Philip N.
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We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ [theta] (logn), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 [superscript 3] n), or O(log2 [superscript 2] n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group.
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Automata, languages, and programming.
Demaine, Erik, Mohammadtaghi Hajiaghayi, and Philip Klein. “Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs.” Automata, Languages and Programming. (Lecture notes in computer science, v. 5555) Springer Berlin / Heidelberg, 2009. 328-340. Copyright © 2009, Springer
Author's final manuscript