Parameterized Algorithms for Node Connectivity Augmentation Problems

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 G0 = (V, E0) by a minimum costs edge set J such that G0 ∪ J has higher connectivity than G0. In the k-Out-Connectivity Augmentation (k-OCA) problem, G0 is (k − 1)-out-connected from s and G0 ∪ J should be k-out-connected from s; in the k-Connectivity Augmentation (k-CA) problem G0 is (k − 1)-connected and G0 ∪ 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 ((2, k)-CA) problem where G0 is (k − 1)-edge-connected and G0 ∪ 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.