A Strong Linear Programming Relaxation for Weighted Tree Augmentation
This work provides a significant algorithmic improvement for a fundamental network design problem, advancing the theoretical understanding of approximation bounds.
The paper tackles the Weighted Tree Augmentation Problem (WTAP) by developing a randomized approximation algorithm that achieves an approximation ratio below 1.49, improving upon the previous state-of-the-art of 1.5+ε.
The Weighted Tree Augmentation Problem (WTAP) is a fundamental network design problem where the goal is to find a minimum-cost set of additional edges (links) to make an input tree 2-edge-connected. While a 2-approximation is standard and the integrality gap of the classic Cut LP relaxation is known to be at least 1.5, achieving approximation factors significantly below 2 has proven challenging. Recent advances of Traub and Zenklusen using local search culminated in a ratio of $1.5+ε$, establishing the state-of-the-art. In this work, we present a randomized approximation algorithm for WTAP with an approximation ratio below 1.49. Our approach is based on designing and rounding a strong linear programming relaxation for WTAP which incorporates variables that represent subsets of edges and the links used to cover them, inspired by lift-and-project methods like Sherali-Adams.