Kernelization and complexity results for connectivity augmentation problems

Abstract

Connectivity augmentation problems ask for adding a set of at most k edges (called links) whose insertion makes a given graph satisfy a specified connectivity property, such as bridge-connectivity or biconnectivity. A bridge-connected (biconnected) graph is a connected graph that does not possess an edge (a vertex) whose removal results in a disconnected graph. We show that, for bridge-connectivity and biconnectivity, the respective connectivity augmentation problems admit problem kernels with O(k 2 ) vertices and links. Moreover, we study partial connectivity augmentation problems, naturally generalizing connectivity augmentation problems. Here, we do not require that, after adding the edges, the entire graph should satisfy the connectivity property, but a large subgraph. In this setting, three polynomial-time solvable connectivity augmentation problems behave differently, namely, the partial bridge-connectivity augmentation problem and the partial biconnectivity augmentation problem remain polynomial-time solvable whereas the partial strong connectivity augmentation problem becomes W[2]-hard with respect to k.

Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

“…Graphs in this paper are assumed to be undirected and may have parallel edges, unless stated otherwise. While there was a large progress in the study of parameterized complexity of edge-connectivity problems [22,28,5,1,16], many papers mention that very little is known about their much harder node-connectivity counterparts. We will consider the "simplest" type of node-connectivity problems, that however have a rich history, when the goal is to increase the node connectivity from k − 1 to k from a given node to other nodes, or between all nodes.…”

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

“…Arguments of this form cannot be generalized to the case when the links have different costs, as the more useful links can have higher costs. Our results go beyond the results of [21,12] by considering higher order edge-connectivity and by allowing arbitrary costs on the links.…”

“…Assuming that the number p of new links is much smaller than the size of the graph, exponential dependence on p is still acceptable, as long as the running time depends only polynomially on the size of the graph. It follows from Nagamochi [21,Lemma 7] that Minimum Cardinality Edge-Connectivity Augmentation from 1 to 2 is fixed-parameter tractable parameterized by this upper bound p. Guo and Uhlmann [12] showed that this problem, as well as its node-connectivity counterpart, admits a kernel of O(p 2 ) nodes and O(p 2 ) links. Neither of these algorithms seem to work for the more general minimum cost version of the problem, as the algorithms rely on discarding links that can be replaced by more useful ones.…”

A survey of parameterized algorithms and the complexity of edge modification

“…The first parameterized algorithm for the connectivity augmentation problem was considered by Nagamochi [224], who gave a 2 O(k log k) |V | O(1) algorithm for the case when the weights on the links are identical and λ is odd. Guo and Uhlmann [158] gave a kernel with O(k 2 ) vertices and links for the same case. Marx and Vegh [216] studied the problem in its full generality and gave a kernel with O(k) vertices, O(k 3 ) links and weights of (k 6 log k) bit integers.…”