A Strong Linear Programming Relaxation for Weighted Tree Augmentation

2026-03-31

AI Summary

This work addresses the Weighted Tree Augmentation Problem (WTAP), which seeks to add edges of minimum total cost to a given tree so that the resulting graph becomes 2-edge-connected. The authors introduce, for the first time, a higher-order linear programming relaxation inspired by the Sherali-Adams hierarchy for WTAP, formulating a novel variable system based on subsets of edges and their covering relationships. By integrating lift-and-project techniques with randomized rounding, their approach breaks through the long-standing 1.5-approximation barrier, achieving an approximation ratio strictly below 1.49. This result improves upon the previous best-known guarantee of 1.5 + ε, marking a significant advance in the approximability of WTAP.

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 $1.5+ε$, 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.

Problem

• Weighted Tree Augmentation
• 2-edge-connected
• network design
• approximation algorithm
• linear programming relaxation

Innovation

• Weighted Tree Augmentation Problem
• strong linear programming relaxation
• lift-and-project
• approximation algorithm
• randomized rounding