Abstract
In the Tree Augmentation problem we are given a tree $$T=(V,F)$$ and a set $$E \subseteq V \times V$$ of edges with positive integer costs $$\{c_e:e \in E\}$$ . The goal is to augment T by a minimum cost edge set $$J \subseteq E$$ such that $$T \cup J$$ is 2-edge-connected. We obtain the following results.
Highlights
We consider the following problem: Tree Augmentation Input: A tree T = (V, F ) and an additional set E ⊆ V × V of edges with positive integer costs c = {ce : e ∈ E}
It is known that the integrality gap of a standard LP-relaxation for the problem, so called Cut-LP, is at most 2 [11] and at least 1.5 [4]
Several other LP and SDP relaxations were introduced to show that the algorithm in [8, 9, 17] achieves ratio better than 2 w.r.t. these relaxations, cf. [2, 16]
Summary
For any 1 ≤ λ ≤ k − 1, Tree Augmentation admits a 4k · poly(n) time algorithm that computes a solution of cost at most ρ+. We repeatedly take a certain complete rooted subtree S, and either find a k-branch-constraint violated by some branch in S, or a “cheap” cover JS of a subset S of the T -edges of S; in the latter case, we add JS to our partial solution J, contract S, and iterate on the instance T ← T /S. Any polynomial time algorithm that computes a solution J of cost at most 2 times the optimal value of the Cut-LP for covering F has the desired property. There exists a 4k · poly(n) time algorithm that either finds a k-branch constraint violated by x, or computes a solution of cost ≤ ρ e∈E\R cexe. It only remains to prove Lemma 4, which we will do in the subsequent sections
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