Improved Approximation for Weighted Tree Augmentation with Bounded Costs

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 ⊆ V × V disjoint from E called links, and a cost vector c ∈ R≥0, WTAP asks to find a minimum-cost set F ⊆ L with the property that (V,E ∪ F ) is 2-edge connected. The special case where c` = 1 for all ` ∈ 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 ∈ Z≥1 and > 0, we present an LP based (δ+ )-approximation for WTAP restricted to cost vectors c in [1,M ] for δ ≈ 1.96417. For the special case of TAP we improve this factor to 53 + . 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 β-simple pairs losing only an -factor in the objective function. We then show how to use the additional constraints in our LP combined with the β-simple structure to round a fractional solution in each part of the decomposition. ar X iv :1 60 7. 03 79 1v 2 [ cs .D S] 1 5 Se p 20 16