The primal-dual method for approximation algorithms and its application to network design problems
Approximation algorithms for NP-hard problems
Approximation algorithms for finding highly connected subgraphs
Approximation algorithms for NP-hard problems
Discrete Applied Mathematics - Special issue: Special issue devoted to the fifth annual international computing and combinatories conference (COCOON'99) Tokyo, Japan 26-28 July 1999
On 2-Coverings and 2-Packings of Laminar Families
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
A 1.8 approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2
ACM Transactions on Algorithms (TALG)
On the integrality ratio for tree augmentation
Operations Research Letters
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
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The Tree Augmentation Problem (TAP) is: given a tree T=(V,E) and a set E of edges (called links) on V disjoint to E, find a minimum-size edge-subset F@?E such that T+F is 2-edge-connected. TAP is equivalent to the problem of finding a minimum-size edge-cover F@?E of a laminar set-family. We consider the restriction, denoted LL-TAP, of TAP to instances when every link in E connects two leaves of T. The best approximation ratio for TAP is 3/2, obtained by Even et al. (2001, 2009, 2008) [3-5], and no better ratio was known for LL-TAP. All the previous approximation algorithms that achieve a ratio better than 2 for TAP, or even for LL-TAP, have been quite involved. For LL-TAP we obtain the following approximation ratios: 17/12 for general trees, 11/8 for trees of height 3, and 4/3 for trees of height 2. We also give a very simple3/2-approximation algorithm (for general trees) and prove that it computes a solution of size at most min{32t,53t^*}, where t is the minimum size of an edge-cover of the leaves, and t^* is the optimal value of the natural LP-relaxation for the problem of covering the leaf edges only. This provides the first evidence that the integrality gap of a natural LP-relaxation for LL-TAP is less than 2.