Approximating weighted tree augmentation via Chvátal-Gomory cuts

The weighted tree augmentation problem (WTAP) is a fundamental network design problem. We are given an undirected tree G = (V;E) with n = jV j nodes, an additional set of edges L called links and a cost vector c ϵ RL>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 cℓ = 1 for all ℓ ϵ 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 ϵ for WTAP by (Frederickson and JáJá, '81) and 3 ϵ for TAP due to (Kortsarz and Nutov, TALG '16). Adjashvili (SODA '17) recently presented an 1:96418 + "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 ϵ 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 ϵ + "-approximation for WTAP under the bounded cost assumption. We achieve this by introducing a strong LP that combines 0; 1 ϵ -ChvátalGomory 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 guarantee of 3 2+", 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.