Parameterized algorithms for node connectivity augmentation problems
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A graph G is k-out-connected from its node s if it contains k internally disjoint sv-paths to every node v; G is k-connected if it is k-out-connected from every node. In connectivity augmentation problems the goal is to augment a graph G_0=(V,E_0) by a minimum costs edge set J such that G_0 ∪ J has higher connectivity than G_0. In the k-Out-Connectivity Augmentation (k-OCA) problem, G_0 is (k-1)-out-connected from s and G_0 ∪ J should be k-out-connected from s; in the k-Connectivity Augmentation (k-CA) problem G_0 is (k-1)-connected and G_0 ∪ J should be k-connected. The parameterized complexity status of these problems was open even for k=3 and unit costs. We will show that k-OCA and 3-CA can be solved in time 9^p · n^O(1), where p is the size of an optimal solution. Our paper is the first that shows fixed parameter tractability of a k-node-connectivity augmentation problem with high values of k. We will also consider the (2,k)-Connectivity Augmentation problem where G_0 is (k-1)-edge-connected and G_0 ∪ J should be both k-edge-connected and 2-connected. We will show that this problem can be solved in time 9^p · n^O(1), and for unit costs approximated within 1.892.
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