We study the Weighted Tree Augmentation Problem for general link costs. We show that the integrality gap of the ODD-LP relaxation for the (weighted) Tree Augmentation Problem for a $k$-level tree instance is at most $2 - \frac{1}{2^{k-1}}$. For 2- and 3-level trees, these ratios are $\frac32$ and $\frac74$ respectively. Our proofs are constructive and yield polynomial-time approximation algorithms with matching guarantees.

翻译：我们研究了“加权树增殖问题”的一般连结成本。我们证明“ODD-LP”对(加权的)树增殖问题的“ODD-LP”整体性差距,对于一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵一棵