Path-contractions, edge deletions and connectivity preservation

Abstract

We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: (Weighted) Biconnectivity Deletion, where we are tasked with deleting k edges while preserving biconnectivity in an undirected graph, Vertex-deletion Preserving Strong Connectivity, where we want to maintain strong connectivity of a digraph while deleting exactly k vertices, and Path-contraction Preserving Strong Connectivity, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are W[1]-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable and we provide a 2 O(k log k) n O(1) -algorithm that solves Weighted Biconnectivity Deletion. 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].

Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

“…For example, Bang-Jensen et al [4] show that the k-Edge Connected Subgraph problem is fixed parameter tractable for the combined parameter of k and the size of a deletion set. Gutin et al [23] show a similar result for k-Connected Subgraph with unit costs.…”

“…Corollary 2 resolves the open question of parameterized complexity of 3-CA [28,23], by establishing that 3-CA is fixed parameter tractable.…”

“…The reduction of Basavaraju et al [5] was generalized in [36] to the problem of covering a crossing set family by edges (see definitions below), and also was extended to 2-CA; specifically, the result of [36] implies that 2-CA and 2-OCA can also be solved in time 9 p • n O (1) . However, the parameterized complexity status of both 3-OCA and 3-CA was open even for unit costs [28,23].…”

The Parameterized Complexity of the Survivable Network Design Problem

“…Bang-Jensen et al [5] show that this problem is FPT for the combined parameter of d and the size of a deletion set, i.e., the number of edges to be removed from the input graph in order to obtain a minimum cost solution. For the same parametrization, Gutin et al [38] provide a (non-uniform) FPT algorithm for the vertex-connectivity variant called d-Vertex Connected Subgraph on unweighted graphs. The authors of [5] also note that requiring a spanning solution H (i.e., when d s,t ≥ 1 for all s, t ∈ V ) makes SNDP trivially FPT when parameterizing by the solution size (i.e, the number of edges of the solution H), since any spanning subgraph…”

The Parameterized Complexity of the Survivable Network Design Problem

“…Bang-Jensen et al [5] show that this problem is FPT for the combined parameter of d and the size of a deletion set, i.e., the number of edges to be removed from the input graph in order to obtain a minimum cost solution. For the same parametrization, Gutin et al [38] provide a (non-uniform) FPT algorithm for the vertex-connectivity variant called d-Vertex Connected Subgraph on unweighted graphs. The authors of [5] also note that requiring a spanning solution H (i.e., when d s,t ≥ 1 for all s, t ∈ V ) makes SNDP trivially FPT when parameterizing by the solution size (i.e, the number of edges of the solution H), since any spanning subgraph has at least n − 1 edges.…”