Local Search for Weighted Tree Augmentation and Steiner Tree

TL;DR

This paper introduces a non-oblivious local search technique that improves approximation algorithms for Weighted Tree Augmentation and Steiner Tree, achieving better ratios without LP reliance.

Contribution

The authors develop a purely combinatorial local search method that enhances approximation ratios for Weighted Tree Augmentation and Steiner Tree problems.

Findings

Achieves a (1.5+ε)-approximation for Weighted Tree Augmentation.

Provides an alternative proof for the Steiner Tree approximation factor of ln 4 + ε.

Eliminates the need for LP solving in approximation algorithms, simplifying the approach.

Abstract

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 $(1.5+\epsilon)$-approximation for Weighted Tree Augmentation, improving on a recent relative greedy based method with approximation factor $1+\ln 2 + \epsilon\approx 1.69$. 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 $\ln 4 + \epsilon$. 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 $\ln 4$.