A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

Abstract

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusionminimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a ( k 2 + 1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V | 3 |E|) = O(|V | 5 ). * Up to 1990,

Approximating Minimum-Power k-Connectivity

“…Combining Theorem 2 with a direct analysis of the algorithms in [2,6,14] for MCkCS, we obtain the following result: Theorem 3. Suppose that MPkIS admits a γ-approximation algorithm and that MPkDP admits a θ-approximation algorithm.…”

“…The fact that the returned graph is k-connected was already established in [2,14], and easily follows from the definition of F . For any constant k, Step 1 can be implemented in polynomial time, by Lemma 3.…”

“…Halin [10] proved that any minimally k-connected graph has a node of degree k. A stronger statement was proved by Mader [19]: This motivates the following auxiliary problem, which min-cost variant is the basis for the algorithms in [2,6,14].…”

“…As was mentioned, currently β = min{k, O(log n)} [9], and β = 3/2 for k = 2 [13] (note that here β is the ratio for MP(k − 1)EC and not for MPkEC). The best known values for α are: α = (k + 1)/2 for 2 ≤ k ≤ 7 (see [2] for k = 2, 3, [6] for k = 4, 5, and [14] for k = 6, 7), α = k for other small values of k [14], and α = O log k · log n n−k otherwise [20]. Thus for undirected MPkCS the following ratios follow: 3k for any k, k + 2 (k + 1)/2 for 2 ≤ k ≤ 7, O(log n) unless k = n − o(n), and O(log 2 n) if k = n − o(n).…”

“…For results on min-cost k-connectivity problems see [2,6,14,5,15,7,20,18]; see also a recent survey in [16] on various min-cost connectivity problems.…”