Approximating {0,1,2}-Survivable Networks with Minimum Number of Steiner Points

We consider low connectivity variants of the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set R of terminals in a metric space (M,d), a subset B ⊆ R of “unstable” terminals, and connectivity requirements {ruv : u, v ∈ R}, find a minimum size set S ⊆ M of additional points such that the unit-disc graph of R∪S contains ruv pairwise internally edge-disjoint and (B ∪ S)-disjoint uv-paths for all u, v ∈ R. The case when ruv = 1 for all u, v ∈ R is the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem, and the case ruv ∈ {0, 1} is the Steiner Forest with Minimum Number of Steiner Points (SF-MSP) problem. Let ∆ be the maximum number of points in a unit ball such that the distance between any two of them is larger than 1. It is known that ∆ = 5 in R. The previous known approximation ratio for ST-MSP was ⌊(∆ + 1)/2⌋+ 1 + ǫ in an arbitrary normed space [17], and 2.5 + ǫ in the Euclidean space R [5]. Our approximation ratio for ST-MSP is 1 + ln(∆ − 1) + ǫ in an arbitrary normed space, which in R reduces to 1 + ln 4 + ǫ < 2.3863 + ǫ. For SN-MSP with ruv ∈ {0, 1, 2}, we give a simple ∆-approximation algorithm. In particular, for SF-MSP, this improves the previous ratio 2∆. 1 Problems considered A large research effort is focused on developing algorithms for finding a “cheap” network that satisfies a certain property. In wired networks, where connecting any two nodes incurs a cost, many problems can be cast as finding a subgraph of minimum cost that satisfies some prescribed connectivity requirements. Following previous work on min-cost connectivity problems, we use the following generic notion of connectivity. Definition 1.1 Let G = (V,E) be a graph and let Q ⊆ V . The Q-connectivity λQG(u, v) of u, v in G is the maximum number of pairwise (E ∪ Q \ {u, v})-disjoint uv-paths in G. Given connectivity requirements r = {ruv : u, v ∈ R ⊆ V } on a subset R ⊆ V of terminals, we denote by Dr = {uv : u, v ∈ R, ruv > 0} the set of “demand edges” of r. We say that G is (r,Q)-connected, or simply r-connected if Q is understood, if λQG(u, v) ≥ ruv for all uv ∈ Dr. Note that edge-connectivity is the case Q = ∅ and node-connectivity is the case Q = V . The members of E ∪ Q will be called elements, hence λQG(u, v) is the maximum number of pairwise internally element-disjoint uv-paths in G. Variants of the following classic problem were extensively studied in the literature. Survivable Network (SN) Instance: A graph G = (V,E) with edge costs, Q ⊆ V , and connectivity requirements r = {ruv : uv ∈ R ⊆ V }. Objective: Find a minimum-cost (r,Q)-connected subgraph H of G. In practical networks the connectivity requirements are rather small, usually ruv ∈ {0, 1, 2} – so called {0, 1, 2}-SN. Particular cases in this setting are Minimum Spanning Tree (MST) (ruv = 1 for all u, v ∈ V ), Steiner Tree (ruv = 1 for all u, v ∈ R) and Steiner Forest (ruv ∈ {0, 1} for all u, v ∈ R), and 2-Connected Subgraph (ruv = 2 for all u, v ∈ V ). In wireless networks, the range and the location of the transmitters determines the resulting communication network. We consider adding a minimum number of transmitters such that the resulting communication network is (r,Q)-connected. If the range of the transmitters is fixed, our goal is to add a minimum number of transmitters, and we get the following type of problems. Definition 1.2 Let (M,d) be a metric space and let V ⊆ M . The unit-disk graph of V has node set V and edge set {uv : u, v ∈ V, d(u, v) ≤ 1}. Survivable Network with Minimum Number of Steiner Points (SN-MSP) Instance: A finite set R ⊆ M of terminals in a metric space (M,d), a set B ⊆ R of “unstable” terminals, connectivity requirements {ruv : uv ∈ R}. Objective: Find a minimum size set S ⊆ M such that the unit-disk graph of R ∪ S is (r,Q)connected, where Q = B ∪ S. As in previous work, we will allow to place several points at the same location, and assume that the maximum distance between terminals is polynomial in the number of terminals. 2 Previous work and our results On previous work on high connectivity variants of SN problem we refer the reader to a survey in [15] and here only mention some work relevant to this paper. The Steiner Tree problem was studied