Fixed-Parameter Algorithms for Minimum-Cost Edge-Connectivity Augmentation

Abstract

We consider connectivity-augmentation problems in a setting where each potential new edge has a nonnegative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized by p and admits a polynomial kernel. We also prove the fixed-parameter tractability of increasing edge-connectivity from 0 to 2, and increasing node-connectivity from 1 to 2.

Path-contractions, edge deletions and connectivity preservation

“…Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivity-preservation constraints in the parameterized setting.1 Marx and Végh [13] compare [17] and [8] to [9] and [16] with respect to polynomial-time exact and approximation algorithms. 2 Note that since 1-vertex-connectivity is trivially equivalent to 1-edge-connectivity, the 1-vertexconnectivity case was proved to be FPT by Basavaraju et al [4].…”

“…Nagamochi obtained an FPT algorithm for this problem while Guo and Uhlmann showed that this problem, alongside its vertex-connectivity variant, admits a quadratic kernel. Marx and Végh [13] studied the more general problem of augmenting the edge-connectivity of an undirected graph from λ − 1 to λ, via a minimum set of links that has a total cost of at most k, and obtained an FPT algorithm as well as a polynomial kernel for this problem. Basavaraju et al [3] improved the running time of their algorithm and further showed the fixed-parameter tractability of a dual parameterization of this problem.…”

“…We consider the undirected variant of this problem, however, we aim to preserve the vertex-connectivity instead of edge-connectivity. As noted by Marx and Végh [13], vertexconnectivity variants of parameterized connectivity problems seem to be much harder to approach than their edge-connectivity counterparts. 1 Moreover, even the complexity of the problem of augmenting the vertex-connectivity of an undirected graph from 2 to 3, via a minimum set of up to k new links remains open [13].…”

“…1 Marx and Végh [13] compare [17] and [8] to [9] and [16] with respect to polynomial-time exact and approximation algorithms. 2 Note that since 1-vertex-connectivity is trivially equivalent to 1-edge-connectivity, the 1-vertexconnectivity case was proved to be FPT by Basavaraju et al [4].…”

On the Fixed-Parameter Tractability of the Maximum Connectivity Improvement Problem

“…While graph augmentation problems have been widely investigated (see [3,14]), their study from a parameterized complexity perspective is still at the beginning. Marx and Végh [20] proved that the problem of increasing the edge-connectivity of an undirected graph from k − 1 to k by adding at most B edges is fixed-parameter tractable when parameterized by B. They proposed an algorithm with complexity 2 O(B log B) |V | O (1) .…”