Improved Approximation for Weighted Tree Augmentation with Bounded Costs

TL;DR

This paper introduces a new LP-based approximation algorithm for the Weighted Tree Augmentation Problem with bounded costs, achieving improved approximation ratios over the past three decades.

Contribution

It presents the first improved approximation algorithm for WTAP with bounded costs in over thirty years, using a novel LP and a two-phase rounding technique.

Findings

Achieves a $( ext{approximately }1.96417+ ext{epsilon})$-approximation for WTAP with bounded costs.

Improves the approximation ratio for TAP to $rac{5}{3}+ ext{epsilon}$.

Introduces a new LP formulation that differs from existing ones, enabling better bounds.

Abstract

The Weighted Tree Augmentation Problem (WTAP) is a fundamental well-studied problem in the field of network design. Given an undirected tree $G=(V,E)$, an additional set of edges $L \subseteq V\times V$ disjoint from $E$ called \textit{links}, and a cost vector $c\in \mathbb{R}_{\geq 0}^L$, WTAP asks to find a minimum-cost set $F\subseteq L$ with the property that $(V,E\cup F)$ is $2$-edge connected. The special case where $c_\ell = 1$ for all $\ell\in L$ is called the Tree Augmentation Problem (TAP). Both problems are known to be NP-hard. For the class of bounded cost vectors, we present a first improved approximation algorithm for WTAP since more than three decades. Concretely, for any $M\in \mathbb{Z}_{\geq 1}$ and $\epsilon > 0,$ we present an LP based $(\delta+\epsilon)$-approximation for WTAP restricted to cost vectors $c$ in $[1,M]^L$ for $\delta \approx 1.96417$. For the special case of TAP we improve this factor to $\frac{5}{3}+\epsilon$. Our results rely on a new LP, that significantly differs from existing LPs achieving improved bounds for TAP. We round a fractional solution in two phases. The first phase uses the fractional solution to decompose the tree and its fractional solution into so-called $\beta$-simple pairs losing only an $\epsilon$-factor in the objective function. We then show how to use the additional constraints in our LP combined with the $\beta$-simple structure to round a fractional solution in each part of the decomposition.