2-node-connectivity network design
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We consider network design problems in which we are given a graph G=(V,E) with edge costs, and seek a min-cost/size 2-node-connected subgraph G'=(V',E') that satisfies a prescribed property. In the Block-Tree Augmentation problem the goal is to augment a tree T by a min-size edge set F ⊆ E such that G'=T ∪ F is 2-node-connected. We break the natural ratio of 2 for this problem and show that it admits approximation ratio 1.91. This result extends to the related Crossing Family Augmentation problem. In the 2-Connected Dominating Set problem G' should dominate V. We give the first non-trivial approximation algorithm for this problem, with expected ratio Õ(log^4 |V|). In the 2-Connected Quota Subgraph problem we are given node profits p(v) and G' should have profit at least a given quota Q. We show expected ratio Õ(log^2|V|), almost matching the best known ratio O(log^2|V|). Our algorithms are very simple, and they combine three main ingredients. 1. A probabilistic spanning tree embedding with distortion Õ(log |V|) results in a variant of the Block-Tree Augmentation problem. 2. An approximation ratio preserving reduction of Block-Tree Augmentation variants to Node Weighted Steiner Tree problems. 3. Using existing approximation algorithms for variants of the Node Weighted Steiner Tree problem.
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