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.

Two fixed-parameter algorithms for Vertex Covering by Paths on Trees

“…search tree for unweighted VCPT. Finally, pointing to related work [3,9] for the Tree Augmentation problem, we indicate the existence of a size-O(k 2 ) problem kernel for unweighted VCPT.…”

“…Together with five further rules, one of them being fairly technical and dealing with the reduction of degree-two paths, one can show that the size of a then reduced instance can be upper-bounded by a quadratic function exclusively depending on the parameter value k. We omit any details here because of the quite technical proof which, however, works in analogy to a corresponding "kernelization" result for the Tree Augmentation problem [3,9]. Taking for granted that a size-O(k 2 ) problem kernel exists, the "interleaving" of branching and data reduction [7] gives the following result.…”

Two Edge Modification Problems without Polynomial Kernels

“…In recent years the kernelizability of edge modification problems has received considerable attention [5,9,[13][14][15]. For a graph property Π the Π Edge Completion/Editing/Deletion problem is defined as Input: A graph G = (V, E) and an integer k. Parameter: k. Task: Decide whether adding and/or deleting at most k edges in G yields a graph with property Π.…”

“…a disjoint union of cliques. In practice though, the obtained graph will [15] None finite Table 1. Kernelization bounds for some edge modification problems.…”

Fixed-Parameter Algorithms for Minimum Cost Edge-Connectivity Augmentation

“…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 [2003, 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 [2010] 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.…”