ICALP 2014: Automata, Languages, and Programming pp 800-811

# Parameterized Algorithms to Preserve Connectivity

• Manu Basavaraju
• Fedor V. Fomin
• Petr Golovach
• Pranabendu Misra
• M. S. Ramanujan
• Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

## 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 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.

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

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## Authors and Affiliations

• Manu Basavaraju
• 1
• Fedor V. Fomin
• 1
• Petr Golovach
• 1
• Pranabendu Misra
• 2
• M. S. Ramanujan
• 1
• Saket Saurabh
• 1
• 2
1. 1.University of BergenNorway
2. 2.The Institute of Mathematical SciencesChennaiIndia