A Strong Linear Programming Relaxation for Weighted Tree Augmentation

TL;DR

This paper introduces a new linear programming relaxation and a randomized algorithm for the Weighted Tree Augmentation Problem, achieving an approximation ratio below 1.49, improving over previous methods.

Contribution

It presents a novel LP relaxation incorporating subset variables and a randomized rounding technique, advancing the approximation ratio for WTAP.

Findings

Achieved an approximation ratio below 1.49 for WTAP.

Developed a strong LP relaxation inspired by lift-and-project methods.

Improved upon the previous best approximation ratio of 1.5+epsilon.

Abstract

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 $1.5+\epsilon$, 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.