Parameterized Algorithms to Preserve Connectivity

Abstract

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.

We obtain the following results.

• An 9^k|V|^O(1) time algorithm for a weighted version of the augmentation problem. This improves over the previous best bound of 2^O(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 2^O(klogk)|V|^O(1).

• An 2^O(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(k ²) vertices and O(k ²) links when G has even-connectivity.

Our results are based on a novel connection between augmenting sets and the Steiner Tree problem in an appropriately defined auxiliary graph.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 267959, from Bergen Research Foundation and University of Bergen under project BEATING HARDNESS BY PREPROCESSING n.806572, and from PARAPPROX, ERC starting grant n. 306992.