Paper by Erik D. Demaine

Reference:

Erik D. Demaine, MohammadTaghi Hajiaghayi, and Philip Klein, “Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs”, in Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP 2009), Lecture Notes in Computer Science, volume 5555, Rhodes, Greece, July 5–12, 2009, pages 328–340.

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 Θ(log n), 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.

Comments:

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Copyright:

Copyright held by the authors.

Length:

The paper is 12 pages.

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