Node-Connectivity Survivable Network Problems

We survey approximation algorithms and hardness of approximation results for the Survivable Network problem in which we seek a low edge-cost directed/undirected subgraph that satisfies prescribed connectivity demands w.r.t. a given “connectivity measure”. These problems include the following well known problems: Minimum Spanning Tree, Traveling Salesman Problem, Steiner Tree, Steiner Forest, and their directed variants. These are examples of low connectivity Survivable Network problems. In this survey we will consider high connectivity Survivable Network problems; some examples are Min-Cost k-Flow, k-Inconnected Subgraph, kConnected Subgraph, and Rooted Survivable Network. See previous surveys on such problem in [30, 26]. Many common connectivity measures can be defined by the following unified framework. Let G = (V,E) be a (possibly directed) graph. For Q ⊆ V , the Q-connectivity λQG(s, t) of a node pair (s, t) is the maximum number of st-paths such that no two of them have an edge or a node in Q \ {s, t} in common. The case Q = ∅ is the case of edge-connectivity, and we use the notation λG(s, t) := λ ∅ G(s, t); the case Q = V is the case of node-connectivity, and we use the notation κG(s, t) := λ V G(s, t). Even more generally, given node capacities {qv : v ∈ V }, the q-connectivity λ q G(s, t) is the maximum number of pairwise edge disjoint st-paths such that for every v ∈ V \{s, t} at most qv of the paths contain v; Q-connectivity is the particular case when qv ∈ {1,∞} for all v ∈ V and Q = {v ∈ V : qv = 1}. We will consider mainly the nodeconnectivity case Q = V , when the paths are required to be pairwise internally node disjoint. However, most algorithms presented can be adjusted to the q-connectivity case with the same approximation ratio.