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We show that the standard linear programming relaxation for the tree augmentation problem in undirected graphs has an integrality ratio that approaches 3 2 . This refutes a conjecture of Cheriyan, Jordán, and Ravi (ESA 1999) that the integrality ratio is

The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple, deterministic, and runs in time O(kjEj 2 ). The analyses of the heuristics for minimum-size k-node connected spanning subgraphs hinge on theorems of Mader.For undirected graphs and k = 2, a (deterministic) parallel NC version of the heuristic nds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of (1:5 + ) o f m i n i m um, where > 0 is a constant.

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Abstract. We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k 2 . The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation.Key words. approximation algorithm, -critically k-connected graph, linear programming relaxation, network design, k-outconnected graph, vertex connectivity AMS subject classifications. Primary, 68W25, 05C40; Secondary, 68R10, 90C27 DOI. 10.1137/S00975397013922871. Introduction. Let G = (V, E) be an undirected graph, let each edge e ∈ E have a nonnegative cost c e , and let k be a positive integer. The minimum-cost kvertex connected spanning subgraph (mincost k-VCSS) problem is to find a spanning subgraph H of minimum cost such that H is k-vertex connected. (A graph is called k-vertex connected if it has at least k +1 vertices, and the removal of any k −1 vertices leaves a connected graph.) The problem is NP-hard for k ≥ 2, and for k = 1 it is the minimum spanning tree problem. Our paper addresses the "special case" of the problem where the graph has order |V | ≥ 6k 2 ; this too is NP-hard for k ≥ 2. (So for a fixed k, our method handles all graphs except a finite set of "small" graphs, and our method fails on each of the "small" graphs.) Our approximation guarantee is six times the kth harmonic number, which is O(log k). Also, this is an upper bound on the integrality ratio for a standard linear programming relaxation. Several previous papers have attacked the mincost k-VCSS problem (without restrictions on |V |), with the goal of improving on the approximation guarantee (see the references). An approximation guarantee of more than k/2 has been presented in [11]; also, an upper bound of O(k) on the integrality ratio was known [4,5]. Better results were not known for our "special case," but we mention that our results may not improve on previous results for small k (k = 2, 3, 4, . . .). (An O(log k) approximation guarantee was claimed earlier in [15], but subsequently an error has been found, and that claim has been withdrawn; see [16].) For more discussion on related problems and results, see the introduction of [3].Our algorithm is based on two results: (1) a polynomial-time algorithm of Frank and Tardos [5] for finding a minimum-cost k-outconnected subdigraph of a digraph