The Steiner tree problem is one of the most fundamental NP -hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP relaxation of Steiner tree with integrality gap smaller than 2 [Rajagopalan and Vazirani 1999]. In this article we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + ε < 1.39 times the cost of an optimal Steiner tree. The algorithm can be derandomized using the method of limited independence. As a by-product of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering the mentioned open question.

The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani, .In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directedcomponent cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4) + ε < 1.39 approximation for Steiner tree.As a byproduct of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering to the * Extended abstract.

The weighted tree augmentation problem (WTAP) is a fundamental network design problem. We are given an undirected tree G = (V, E) with n = |V | nodes, an additional set of edges L called links and a cost vector c ∈ R L ≥1 . The goal is to choose a minimum cost subset S ⊆ L such that G = (V, E ∪ S) is 2-edgeconnected. In the unweighted case, that is, when we have c = 1 for all ∈ L, the problem is called the tree augmentation problem (TAP).Both problems are known to be APX-hard, and the best known approximation factors are 2 for WTAP by (Frederickson and JáJá, '81) and 3 2 for TAP due to (Kortsarz and Nutov, TALG '16). Adjashvili (SODA '17) recently presented an ≈ 1.96418 + ε-approximation algorithm for WTAP for the case where all link costs are bounded by a constant. This is the first approximation with a better guarantee than 2 that does not require restrictions on the structure of the tree or the links.In this paper, we improve Adjiashvili's approximation to a 3 2 + ε-approximation for WTAP under the bounded cost assumption. We achieve this by introducing a strong LP that combines 0, 1 2 -Chvátal-Gomory cuts for the standard LP for the problem with bundle constraints from Adjiashvili. We show that our LP can be solved efficiently and that it is exact for some instances that arise at the core of Adjiashvili's approach. This results in the improved performance * Martin Groß, Jochen Könemann and Laura Sanità acknowledge support from the NSERC Discovery Grant Program as well as an Early Researcher Award by the Province of Ontario. Samuel Fiorini acknowledges support from F.R.S. -FNRS (grant number 1861993, "Polyhedral combinatorics: cycle transversals, {0, 1/2}-cuts") and the ERC (Consolidator Grant 615640-ForEFront). +ε, which is asymptotically on par with the result by Kortsarz and Nutov. Our result also is the best-known LP-relative approximation algorithm for TAP.

In the node-weighted prize-collecting Steiner tree problem (NW-PCST) we are given an undirected graph G = (V, E), non-negative costs c(v) and penalties π(v) for each v ∈ V . The goal is to find a tree T that minimizes the total cost of the vertices spanned by T plus the total penalty of vertices not in T . This problem is well-known to be set-cover hard to approximate. Moss and Rabani (STOC'01) presented a primal-dual Lagrangean-multiplier-preserving O(ln |V |)-approximation algorithm for this problem. We show a serious problem with the algorithm, and present a new, fundamentally different primal-dual method achieving the same performance guarantee. Our algorithm introduces several novel features to the primal-dual method that may be of independent interest.

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