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• W2952482868 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. Recently, Adjiashvili [SODA 17] introduced a novel LP for the problem and used it to break the $2$-approximation barrier for instances when the maximum cost $M$ of an edge in $E$ is bounded by a constant; his algorithm computes a $1.96418+epsilon$ approximate solution in time $n^{{(M/epsilon^2)}^{O(1)}}$. Using a simpler LP, we achieve ratio $frac{12}{7}+epsilon$ in time $2^{O(M/epsilon^2)} poly(n)$.This gives ratio better than $2$ for logarithmic costs, and not only for constant costs. One of the oldest open questions for the problem is whether for unit costs (when $M=1$) the standard LP-relaxation, so called Cut-LP, has integrality gap less than $2$. We resolve this open question by proving that for unit costs the integrality gap of the Cut-LP is at most $28/15=2-2/15$. In addition, we will prove that another natural LP-relaxation, that is much simpler than the ones in previous work, has integrality gap at most $7/4$." @default.
• W2952482868 created "2019-06-27" @default.
• W2952482868 creator A5003622145 @default.
• W2952482868 date "2017-03-21" @default.
• W2952482868 modified "2023-09-27" @default.
• W2952482868 title "On the Tree Augmentation Problem" @default.
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