Parameterized Algorithms to Preserve Connectivity
We study the following family of connectivity problems. For a given λ-edge connected (multi) graph G = (V,E), a set of links L such that G + L = (V, E ∪ L) is (λ + 1)-edge connected, and a positive integer k, the questions are
Augmentation Problem: whether G can be augmented to a (λ + 1)-edge connected graph by adding at most k links from L; or
Deletion Problem: whether it is possible to preserve (λ + 1)-edge connectivity of graph G + L after deleting at least k links from L.
An 9k|V|O(1) time algorithm for a weighted version of the augmentation problem. This improves over the previous best bound of 2O(klogk)|V|O(1) given by Marx and Vegh [ICALP 2013]. Let us remark that even for λ = 1, the best known algorithm so far due to Nagamochi [DAM 2003] runs in time 2O(klogk)|V|O(1).
An 2O(k)|V|O(1) algorithm for the deletion problem thus establishing that the problem is fixed-parameter tractable (FPT). Moreover, we show that the problem admits a kernel with 12k vertices and 3k links when the graph G has odd-connectivity and a kernel with O(k2) vertices and O(k2) links when G has even-connectivity.
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