Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs

Authors:
Erik D. Demaine;Mohammadtaghi Hajiaghayi;Philip N. Klein
Affiliations:
MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, USA 02139;AT&T Labs -- Research, Florham Park, USA 07932;Department of Computer Science, Brown University, Providence, USA 02912
Venue:
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Year:
2009

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Abstract

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 *** (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 n ), or O (log2 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.