Title: A $\frac{3}{2}$-Approximation Algorithm for Tree Augmentation via Chvátal-Gomory Cuts,
Abstract: The weighted tree augmentation problem (WTAP) is a fundamental network design
problem. We are given an undirected tree $G = (V,E)$, an additional set of
edges $L$ called links and a cost vector $c \in \mathbb{R}^L_{\geq 1}$. The
goal is to choose a minimum cost subset $S \subseteq L$ such that $G = (V, E
\cup S)$ is $2$-edge-connected. In the unweighted case, that is, when we have
$c_\ell = 1$ for all $\ell \in L$, the problem is called the tree augmentation
problem (TAP).
Both problems are known to be APX-hard, and the best known approximation
factors are $2$ for WTAP by (Frederickson and JáJá, '81) and $\tfrac{3}{2}$
for TAP due to (Kortsarz and Nutov, TALG '16). In the case where all link costs
are bounded by a constant $M$, (Adjiashvili, SODA '17) recently gave a $\approx
1.96418+\varepsilon$-approximation algorithm for WTAP under this assumption.
This is the first approximation with a better guarantee than $2$ that does not
require restrictions on the structure of the tree or the links.
In this paper, we improve Adjiashvili's approximation to a
$\frac{3}{2}+\varepsilon$-approximation for WTAP under the bounded cost
assumption. We achieve this by introducing a strong LP that combines
$\{0,\frac{1}{2}\}$-Chvátal-Gomory cuts for the standard LP for the problem
with bundle constraints from Adjiashvili. We show that our LP can be solved
efficiently and that it is exact for some instances that arise at the core of
Adjiashvili's approach. This results in the improved guarantee of
$\frac{3}{2}+\varepsilon$. For TAP, this is the best known LP-based result, and
matches the bound of $\frac{3}{2}+\varepsilon$ achieved by the best SDP-based
algorithm due to (Cheriyan and Gao, arXiv '15).