Local Search for Weighted Tree Augmentation and Steiner Tree

We present a technique that allows for improving on some relative greedy procedures by well-chosen (non-oblivious) local search algorithms. Relative greedy procedures are a particular type of greedy algorithm that start with a simple, though weak, solution, and iteratively replace parts of this starting solution by stronger components. Some well-known applications of relative greedy algorithms include approximation algorithms for Steiner Tree and, more recently, for connectivity augmentation problems. The main application of our technique leads to a (1.5+ε)-approximation for Weighted Tree Augmentation, improving on a recent relative greedy based method with approximation factor 1+ln 2+ε ≈ 1.69. Furthermore, we show how our local search technique can be applied to Steiner Tree, leading to an alternative way to obtain the currently best known approximation factor of ln 4 + ε. Contrary to prior methods, our approach is purely combinatorial without the need to solve an LP. Nevertheless, the solution value can still be bounded in terms of the well-known hypergraphic LP, leading to an alternative, and arguably simpler, technique to bound its integrality gap by ln 4. ∗This project received funding from Swiss National Science Foundation grant 200021 184622 and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 817750). ‡Department of Mathematics, ETH Zurich, Zurich, Switzerland. Email: vera.traub@ifor.math.ethz.ch. §Department of Mathematics, ETH Zurich, Zurich, Switzerland. Email: ricoz@ethz.ch. ar X iv :2 10 7. 07 40 3v 1 [ cs .D S] 1 5 Ju l 2 02 1